1215 lines
275 KiB
HTML
1215 lines
275 KiB
HTML
<!DOCTYPE html><html lang="zh-CN"> <head><meta charset="UTF-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="description" content="这篇文章介绍了三种直线扫描转换算法：DDA算法、中点画线算法和Bresenham算法。 DDA算法通过计算直线起点和终点的坐标差值来确定步数和增量，依次绘制直线上的点。算法实现简单，适用于任意直线，但精度较低。 中点画线算法仅适用于斜率为0或1的直线，通过计算中点和斜率来确定直线上的像素点，效率较高。 Bresenham算法根据直线的斜率和误差项来确定像素点的位置，适用于任意直线，且精度高，但实现稍复杂。"><meta name="author" content="biss"><meta name="keywords" content="技术, 学习, OpenGL"><link rel="canonical" href="https://blog.biss.click/posts/7207243b/"><link rel="alternate" type="application/rss+xml" title="Bi's Blog RSS" href="https://blog.biss.click/rss.xml"><meta property="og:site_name" content="Bi's Blog"><meta property="og:locale" content="zh_CN"><meta property="og:type" content="article"><meta property="og:title" content="OpenGL-直线的扫描转换 | Bi's Blog"><meta property="og:description" content="这篇文章介绍了三种直线扫描转换算法：DDA算法、中点画线算法和Bresenham算法。 DDA算法通过计算直线起点和终点的坐标差值来确定步数和增量，依次绘制直线上的点。算法实现简单，适用于任意直线，但精度较低。 中点画线算法仅适用于斜率为0或1的直线，通过计算中点和斜率来确定直线上的像素点，效率较高。 Bresenham算法根据直线的斜率和误差项来确定像素点的位置，适用于任意直线，且精度高，但实现稍复杂。"><meta property="og:url" content="https://blog.biss.click/posts/7207243b/"><meta property="og:image" content="https://pic.biss.click/image/c5457adc-214c-4a18-9aa6-43fbc0bfc2f4.webp"><meta property="og:image:alt" content="OpenGL-直线的扫描转换"><meta name="twitter:card" content="summary_large_image"><meta name="twitter:title" content="OpenGL-直线的扫描转换 | Bi's Blog"><meta name="twitter:description" content="这篇文章介绍了三种直线扫描转换算法：DDA算法、中点画线算法和Bresenham算法。 DDA算法通过计算直线起点和终点的坐标差值来确定步数和增量，依次绘制直线上的点。算法实现简单，适用于任意直线，但精度较低。 中点画线算法仅适用于斜率为0或1的直线，通过计算中点和斜率来确定直线上的像素点，效率较高。 Bresenham算法根据直线的斜率和误差项来确定像素点的位置，适用于任意直线，且精度高，但实现稍复杂。"><meta name="twitter:image" content="https://pic.biss.click/image/c5457adc-214c-4a18-9aa6-43fbc0bfc2f4.webp"><meta property="article:published_time" content="2026-04-11T19:01:02.000Z"><meta property="article:modified_time" content="2026-04-11T19:01:02.000Z"><meta property="article:author" content="biss"><meta property="article:tag" content="技术"><meta property="article:tag" content="学习"><meta property="article:tag" content="OpenGL"><script type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","headline":"OpenGL-直线的扫描转换","description":"这篇文章介绍了三种直线扫描转换算法：DDA算法、中点画线算法和Bresenham算法。 DDA算法通过计算直线起点和终点的坐标差值来确定步数和增量，依次绘制直线上的点。算法实现简单，适用于任意直线，但精度较低。 中点画线算法仅适用于斜率为0或1的直线，通过计算中点和斜率来确定直线上的像素点，效率较高。 Bresenham算法根据直线的斜率和误差项来确定像素点的位置，适用于任意直线，且精度高，但实现稍复杂。","image":["https://pic.biss.click/image/c5457adc-214c-4a18-9aa6-43fbc0bfc2f4.webp"],"datePublished":"2026-04-11T19:01:02.000Z","dateModified":"2026-04-11T19:01:02.000Z","wordCount":1100,"inLanguage":"zh-CN","mainEntityOfPage":{"@type":"WebPage","@id":"https://blog.biss.click/posts/7207243b/"},"author":{"@type":"Person","name":"biss","url":"https://github.com/bishshi"},"publisher":{"@type":"Person","name":"biss","image":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"keywords":"技术, 学习, OpenGL","articleSection":"技术","isPartOf":{"@type":"CreativeWorkSeries","name":"OpenGL"}}</script><link rel="icon" href="/images/Bi.ico"><link rel="manifest" href="/manifest.webmanifest"><link rel="apple-touch-icon" href="/images/apple-touch-icon.png"><meta name="theme-color" content="#f7f5ef" media="(prefers-color-scheme: light)"><meta name="theme-color" content="#0b1014" media="(prefers-color-scheme: dark)"><script defer src="https://umami.biss.click/script.js" data-website-id="3d97b310-b241-4fc6-bc9c-68e317fc7d42"></script><script>window.SITE_CONFIG = {"site":{"url":"https://blog.biss.click","name":"Bi's Blog","description":"Personal blog by biss.","language":"zh-CN","domain":"blog.biss.click","slogan":"好奇心转化为代码的空间。","startTime":"2025-01-01T00:00:00+08:00","copyrightStartYear":2025,"license":"内容采用 CC BY-NC-SA 4.0 协议"},"author":{"name":"biss","email":"bishsh2006@gmail.com","github":"https://github.com/bishshi","avatar":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"comments":{"envId":"https://comment.biss.click","region":"","script":"https://cdn.jsdelivr.net/npm/twikoo@1.7.7/dist/twikoo.min.js"},"analytics":{"umami":{"script":"https://umami.biss.click/script.js","websiteId":"3d97b310-b241-4fc6-bc9c-68e317fc7d42","statsEndpoint":"https://blogumami.biss.click"}},"talks":{"apiUrl":"https://mm.biss.click/api/echo/page","pageSize":30,"homeLimit":6,"cacheKey":"talksCache","cacheTimeKey":"talksCacheTime","cacheDurationMs":1800000,"fallbackAvatar":"https://free.picui.cn/free/2025/08/10/689845496a283.png"},"search":{"apiKey":"2uxQzk7eD2GBCljdSmIiuU3ause7UK9n","server":{"host":"typesense.biss.click","port":"443","protocol":"https"},"indexName":"blogs","searchParams":{"query_by":"title,content","highlight_full_fields":"title,content","per_page":8,"num_typos":1,"typo_tokens_threshold":1,"prefix":true},"ui":{"maxRetries":10,"retryDelay":100,"animationDuration":300,"maxInitRetries":30,"placeholder":"输入关键词寻找故事..."}},"externalLinks":{"waitSeconds":10,"whitelist":["blog.biss.click","biss.click","github.com","localhost","127.0.0.1"]}};</script><script>
|
||
(() => {
|
||
const storageKey = 'site-theme';
|
||
const media = window.matchMedia('(prefers-color-scheme: dark)');
|
||
const getSystemTheme = () => (media.matches ? 'dark' : 'light');
|
||
const getSavedTheme = () => {
|
||
try {
|
||
const theme = localStorage.getItem(storageKey);
|
||
return theme === 'dark' || theme === 'light' ? theme : null;
|
||
} catch {
|
||
return null;
|
||
}
|
||
};
|
||
|
||
document.documentElement.dataset.theme = getSavedTheme() || getSystemTheme();
|
||
})();
|
||
</script><link rel="stylesheet" href="https://cdn.jsdmirror.com/npm/instantsearch.css/themes/reset-min.css"><link rel="stylesheet" href="https://cdn.jsdmirror.com/npm/@fancyapps/ui@5.0/dist/fancybox/fancybox.css"><link rel="stylesheet" href="https://cdn.jsdmirror.com/npm/@fortawesome/fontawesome-free@6.7.2/css/all.min.css"><title>OpenGL-直线的扫描转换 | Bi's Blog</title><link rel="stylesheet" href="/_astro/BaseLayout.DvPRLM9I.css"></head> <body> <header class="site-header"> <nav class="nav"> <a class="brand" href="/" aria-label="Bi's Blog 首页"><span>biss</span><i>.log</i></a> <button class="nav-menu-toggle" type="button" aria-label="打开菜单" aria-expanded="false" data-nav-toggle> <i class="fa-solid fa-bars nav-menu-open" aria-hidden="true"></i> <i class="fa-solid fa-xmark nav-menu-close" aria-hidden="true"></i> </button> <div class="nav-drawer-backdrop" data-nav-backdrop></div> <div class="nav-menu" data-nav-menu> <div class="nav-links nav-center"> <a href="/archives/"><i class="fa-solid fa-box-archive nav-icon" aria-hidden="true"></i><span>归档</span></a><a href="/categories/"><i class="fa-solid fa-folder-open nav-icon" aria-hidden="true"></i><span>分类</span></a><a href="/tags/"><i class="fa-solid fa-tags nav-icon" aria-hidden="true"></i><span>标签</span></a><a href="/shuoshuo/"><i class="fa-regular fa-comments nav-icon" aria-hidden="true"></i><span>说说</span></a><a href="/about/"><i class="fa-solid fa-heart nav-icon" aria-hidden="true"></i><span>关于</span></a><a href="/link/"><i class="fa-solid fa-link nav-icon" aria-hidden="true"></i><span>友链</span></a> </div> <div class="nav-links nav-actions"> <button class="nav-random" type="button" onclick="randomPost()" aria-label="前往随机文章" title="前往随机文章"> <i class="fa-solid fa-shuffle nav-icon" aria-hidden="true"></i> <span>随机文章</span> </button> <button class="nav-search search-typesense-trigger" type="button" aria-label="搜索" title="搜索"> <i class="fa-solid fa-magnifying-glass nav-icon" aria-hidden="true"></i> <span>搜索</span> </button> <button class="nav-console" type="button" aria-label="打开中控台" title="中控台" data-console-open> <i class="fa-solid fa-gauge-high nav-icon" aria-hidden="true"></i> <span>中控台</span> </button> <button class="theme-toggle" type="button" aria-label="切换夜间模式" title="切换夜间模式" data-theme-toggle> <i class="fa-solid fa-sun theme-toggle-sun nav-icon" aria-hidden="true"></i> <i class="fa-solid fa-moon theme-toggle-moon nav-icon" aria-hidden="true"></i> </button> </div> </div> </nav> </header> <div class="control-console" data-console-modal aria-hidden="true"> <div class="control-console-backdrop" data-console-close></div> <section class="control-console-panel" role="dialog" aria-modal="true" aria-labelledby="control-console-title" tabindex="-1"> <header class="control-console-head"> <div> <p>CONTROL CENTER</p> <h2 id="control-console-title">中控台</h2> </div> <button class="control-console-close" type="button" aria-label="关闭中控台" title="关闭" data-console-close> <i class="fa-solid fa-xmark" aria-hidden="true"></i> </button> </header> <div class="control-console-grid"> <a class="control-site-card" href="https://biss.click" target="_blank" rel="noreferrer"> <i class="fa-solid fa-house-chimney" aria-hidden="true"></i> <span> <strong>主站入口</strong> <em>个人主页与常用入口集合</em> </span> <small>biss.click</small> </a><a class="control-site-card" href="https://mm.biss.click" target="_blank" rel="noreferrer"> <i class="fa-regular fa-comments" aria-hidden="true"></i> <span> <strong>说说动态</strong> <em>短句、碎片记录和近况更新</em> </span> <small>mm.biss.click</small> </a><a class="control-site-card" href="https://comment.biss.click" target="_blank" rel="noreferrer"> <i class="fa-solid fa-message" aria-hidden="true"></i> <span> <strong>评论系统</strong> <em>Twikoo 评论服务与访客互动</em> </span> <small>comment.biss.click</small> </a><a class="control-site-card" href="https://umami.biss.click" target="_blank" rel="noreferrer"> <i class="fa-solid fa-chart-line" aria-hidden="true"></i> <span> <strong>站点统计</strong> <em>Umami 访问统计与站点数据</em> </span> <small>umami.biss.click</small> </a> </div> <div class="control-console-info" aria-label="站点信息"> <section> <span>当前站点</span> <strong>blog.biss.click</strong> </section> <section> <span>站点作者</span> <strong>biss</strong> </section> <section> <span>运行起点</span> <strong>2025-01-01</strong> </section> </div> </section> </div> <main> <div class="reading-progress" aria-hidden="true"><span data-reading-progress></span></div> <aside class="reading-resume" data-reading-resume hidden aria-live="polite"> <div class="reading-resume-icon" aria-hidden="true"><i class="fa-solid fa-book-open"></i></div> <div> <strong>继续上次阅读？</strong> <p>你上次读到了 <span data-reading-resume-percent>--</span>。</p> </div> <div class="reading-resume-actions"> <button type="button" data-reading-continue>继续阅读</button> <button type="button" data-reading-restart>从头开始</button> </div> </aside> <section class="post-layout"> <article class="article post-article"> <header class="article-header article-header-cover"> <img class="article-header-cover-img" src="https://pic.biss.click/image/c5457adc-214c-4a18-9aa6-43fbc0bfc2f4.webp" alt="" loading="eager" decoding="async"> <div class="article-header-content"> <div class="article-date-line"> <time datetime="2026-04-11T19:01:02.000Z">发布于 2026/04/12</time> </div> <h1>OpenGL-直线的扫描转换</h1> <div class="article-hero-meta" aria-label="文章信息"> <span>发布于 2026/04/12</span> <a href="/categories/tech/">技术</a> <span>总字数：1.1k</span> <span>阅读时长：3 分钟</span> </div> </div> </header> <section class="article-summary-card" aria-label="文章导读"> <div class="article-summary-header"> <span class="article-summary-brand"> <span class="article-summary-icon" aria-hidden="true">↳</span>
|
||
文章导读
|
||
</span> <span class="article-summary-model">摘要由 Qwen 生成</span> </div> <p class="article-summary-content"> <span class="article-summary-text">这篇文章介绍了三种直线扫描转换算法：DDA算法、中点画线算法和Bresenham算法。 DDA算法通过计算直线起点和终点的坐标差值来确定步数和增量，依次绘制直线上的点。算法实现简单，适用于任意直线，但精度较低。 中点画线算法仅适用于斜率为0或1的直线，通过计算中点和斜率来确定直线上的像素点，效率较高。 Bresenham算法根据直线的斜率和误差项来确定像素点的位置，适用于任意直线，且精度高，但实现稍复杂。</span> </p> </section> <section class="post-series" aria-labelledby="post-series-title"> <header class="post-series-head"> <div> <span>ARTICLE SERIES</span> <h2 id="post-series-title">OpenGL 学习笔记</h2> </div> <strong>第 3 / 4 篇</strong> </header> <ol class="post-series-list"> <li> <a href="/posts/c4477b0c/"> <span>01</span> <strong>配置OpenGL环境</strong> </a> </li><li> <a href="/posts/437a5198/"> <span>02</span> <strong>OpenGL-基础程序</strong> </a> </li><li class="is-current"> <a href="/posts/7207243b/" aria-current="page"> <span>03</span> <strong>OpenGL-直线的扫描转换</strong> </a> </li><li> <a href="/posts/9b2ef00a/"> <span>04</span> <strong>OpenGL- Cohen-Sutherland 编码裁剪算法</strong> </a> </li> </ol> <nav class="post-series-nav" aria-label="系列文章导航"> <a href="/posts/437a5198/" rel="prev"><span>上一篇</span><strong>OpenGL-基础程序</strong></a> <a href="/posts/9b2ef00a/" rel="next"><span>下一篇</span><strong>OpenGL- Cohen-Sutherland 编码裁剪算法</strong></a> </nav> </section> <aside class="butterfly-note butterfly-note--warning article-outdated-note" aria-label="文章时效性提示" data-freshness-at="2026-04-11T19:01:02.000Z" data-has-new-version="false" hidden> <i class="fa-solid fa-triangle-exclamation" aria-hidden="true"></i> <div class="butterfly-note__content"> <p>
|
||
注意：本文最后维护于 2026/04/12，距今已超过一年，部分内容可能已经过时，请结合最新资料判断。
|
||
</p> </div> </aside> <div class="prose" data-toc="[{"depth":1,"text":"DDA数值微分线段算法","id":"dda数值微分线段算法"},{"depth":2,"text":"算法简介","id":"算法简介"},{"depth":2,"text":"算法实现","id":"算法实现"},{"depth":1,"text":"中点画线算法","id":"中点画线算法"},{"depth":2,"text":"算法简介","id":"算法简介-1"},{"depth":2,"text":"算法实现","id":"算法实现-1"},{"depth":1,"text":"Bresenham算法","id":"bresenham算法"},{"depth":2,"text":"算法简介","id":"算法简介-2"},{"depth":2,"text":"算法实现","id":"算法实现-2"}]"> <p>这篇文章来介绍直线扫描转换算法</p>
|
||
<h1 id="dda数值微分线段算法">DDA数值微分线段算法</h1>
|
||
<h2 id="算法简介">算法简介</h2>
|
||
<p>数值微分法即DDA法(Digital Differential Analyzer)，是一种基于微分方程来生成直线的方法。在计算机图形学中，并没有线段的概念，而是一个个像素点组成了线段。</p>
|
||
<p>DDA法生成线段的步骤一般如下：</p>
|
||
<ol>
|
||
<li>有了起始点（<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="6.641ex" height="2.161ex" role="img" focusable="false" viewBox="0 -750 2935.1 955" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-1-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-1-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-1-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1-TEX-N-31"></use></g></g><g data-mml-node="mi" transform="translate(1008.6,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="msub" transform="translate(2008.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-1-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-1-TEX-N-31"></use></g></g></g></g></svg></mjx-container>）和终点（<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="6.96ex" height="2.161ex" role="img" focusable="false" viewBox="0 -750 3076.5 955" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-2-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-2-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-2-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-2-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-2-TEX-I-1D45B"></use></g></g><g data-mml-node="mi" transform="translate(1079.3,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="msub" transform="translate(2079.3,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-2-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-2-TEX-I-1D45B"></use></g></g></g></g></svg></mjx-container>）；</li>
|
||
<li><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="30.336ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 13408.3 999" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-3-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-3-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-3-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-3-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-3-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-3-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-3-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-3-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-3-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-3-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-3-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1682.8,0)"><use data-c="3D" xlink:href="#MJX-3-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2738.6,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-3-TEX-N-7C"></use></g><g data-mml-node="msub" transform="translate(3016.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-3-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(605,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-3-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(4318,0)"><use data-c="2212" xlink:href="#MJX-3-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(5318.3,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-3-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-3-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(6326.8,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-3-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(6604.8,0)"><use data-c="2C" xlink:href="#MJX-3-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(7049.5,0)"><use data-c="394" xlink:href="#MJX-3-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(7882.5,0)"><use data-c="1D466" xlink:href="#MJX-3-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(8650.3,0)"><use data-c="3D" xlink:href="#MJX-3-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(9706,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-3-TEX-N-7C"></use></g><g data-mml-node="msub" transform="translate(9984,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-3-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45B" xlink:href="#MJX-3-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(11203.5,0)"><use data-c="2212" xlink:href="#MJX-3-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(12203.7,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-3-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-3-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(13130.3,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-3-TEX-N-7C"></use></g></g></g></svg></mjx-container></li>
|
||
<li>比较<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="3.179ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 1405 727" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-4-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-4-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-4-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-4-TEX-I-1D465"></use></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.993ex" height="2.084ex" role="img" focusable="false" viewBox="0 -716 1323 921" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-5-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-5-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-5-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-5-TEX-I-1D466"></use></g></g></g></svg></mjx-container>的大小；
|
||
steps=<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="3.179ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 1405 727" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-6-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-6-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-6-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-6-TEX-I-1D465"></use></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.993ex" height="2.084ex" role="img" focusable="false" viewBox="0 -716 1323 921" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-7-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-7-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-7-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-7-TEX-I-1D466"></use></g></g></g></svg></mjx-container>中较大者；</li>
|
||
<li><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -1.091ex;" xmlns="http://www.w3.org/2000/svg" width="27.759ex" height="3.334ex" role="img" focusable="false" viewBox="0 -991.2 12269.5 1473.4" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-8-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path id="MJX-8-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-8-TEX-I-1D452" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path id="MJX-8-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-8-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-8-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-8-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-8-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(469,0)"><use data-c="1D461" xlink:href="#MJX-8-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(830,0)"><use data-c="1D452" xlink:href="#MJX-8-TEX-I-1D452"></use></g><g data-mml-node="msub" transform="translate(1296,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-8-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(2564.2,0)"><use data-c="3D" xlink:href="#MJX-8-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(3620,0)"><g data-mml-node="mrow" transform="translate(525.1,394) scale(0.707)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D465" xlink:href="#MJX-8-TEX-I-1D465"></use></g></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(469,0)"><use data-c="1D461" xlink:href="#MJX-8-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(830,0)"><use data-c="1D452" xlink:href="#MJX-8-TEX-I-1D452"></use></g><g data-mml-node="mi" transform="translate(1296,0)"><use data-c="1D45D" xlink:href="#MJX-8-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(1799,0)"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g></g><rect width="1803.7" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(5663.7,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="mi" transform="translate(6663.7,0)"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(7132.7,0)"><use data-c="1D461" xlink:href="#MJX-8-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(7493.7,0)"><use data-c="1D452" xlink:href="#MJX-8-TEX-I-1D452"></use></g><g data-mml-node="msub" transform="translate(7959.7,0)"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-8-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g></g><g data-mml-node="mo" transform="translate(9170,0)"><use data-c="3D" xlink:href="#MJX-8-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(10225.8,0)"><g data-mml-node="mrow" transform="translate(554.1,485) scale(0.707)"><g data-mml-node="mi"><use data-c="394" xlink:href="#MJX-8-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(833,0)"><use data-c="1D466" xlink:href="#MJX-8-TEX-I-1D466"></use></g></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)"><g data-mml-node="mi"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g><g data-mml-node="mi" transform="translate(469,0)"><use data-c="1D461" xlink:href="#MJX-8-TEX-I-1D461"></use></g><g data-mml-node="mi" transform="translate(830,0)"><use data-c="1D452" xlink:href="#MJX-8-TEX-I-1D452"></use></g><g data-mml-node="mi" transform="translate(1296,0)"><use data-c="1D45D" xlink:href="#MJX-8-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(1799,0)"><use data-c="1D460" xlink:href="#MJX-8-TEX-I-1D460"></use></g></g><rect width="1803.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container></li>
|
||
</ol>
|
||
<h2 id="算法实现">算法实现</h2>
|
||
<p>DDA算法实现如下：</p>
|
||
<pre class="astro-code github-dark" style="background-color:#24292e;color:#e1e4e8; overflow-x: auto; white-space: pre-wrap; word-wrap: break-word;" tabindex="0" data-language="cpp"><code><span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <cmath></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <iostream></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <GL/glut.h></span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 窗口宽度和高度</span></span>
|
||
<span class="line"><span style="color:#F97583">const</span><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> WIDTH </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 640</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#F97583">const</span><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> HEIGHT </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 480</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> drawDDALine</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x1</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y1</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x2</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y2</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#F97583"> float</span><span style="color:#E1E4E8"> x </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> x1;</span></span>
|
||
<span class="line"><span style="color:#F97583"> float</span><span style="color:#E1E4E8"> y </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> y1;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 计算差值</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> dx </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> x2 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> x1;</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> dy </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> y2 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> y1;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 确定步数，取 dx 和 dy 中绝对值较大的那个</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> steps </span><span style="color:#F97583">=</span><span style="color:#B392F0"> std</span><span style="color:#E1E4E8">::</span><span style="color:#B392F0">abs</span><span style="color:#E1E4E8">(dx) </span><span style="color:#F97583">></span><span style="color:#B392F0"> std</span><span style="color:#E1E4E8">::</span><span style="color:#B392F0">abs</span><span style="color:#E1E4E8">(dy) </span><span style="color:#F97583">?</span><span style="color:#B392F0"> std</span><span style="color:#E1E4E8">::</span><span style="color:#B392F0">abs</span><span style="color:#E1E4E8">(dx) </span><span style="color:#F97583">:</span><span style="color:#B392F0"> std</span><span style="color:#E1E4E8">::</span><span style="color:#B392F0">abs</span><span style="color:#E1E4E8">(dy);</span><span style="color:#6A737D"> //三元表达式</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 计算每一步的增量</span></span>
|
||
<span class="line"><span style="color:#F97583"> float</span><span style="color:#E1E4E8"> xIncrement </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> (</span><span style="color:#F97583">float</span><span style="color:#E1E4E8">)dx </span><span style="color:#F97583">/</span><span style="color:#E1E4E8"> steps;</span></span>
|
||
<span class="line"><span style="color:#F97583"> float</span><span style="color:#E1E4E8"> yIncrement </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> (</span><span style="color:#F97583">float</span><span style="color:#E1E4E8">)dy </span><span style="color:#F97583">/</span><span style="color:#E1E4E8"> steps;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 开始绘制点</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glBegin</span><span style="color:#E1E4E8">(GL_POINTS);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glVertex2i</span><span style="color:#E1E4E8">((</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(x), (</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(y));</span><span style="color:#6A737D"> // 绘制起点</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> for</span><span style="color:#E1E4E8"> (</span><span style="color:#F97583">int</span><span style="color:#E1E4E8"> k </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 0</span><span style="color:#E1E4E8">; k </span><span style="color:#F97583"><</span><span style="color:#E1E4E8"> steps; k</span><span style="color:#F97583">++</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> x </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> xIncrement;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> y </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> yIncrement;</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 将浮点坐标四舍五入取整转换为整数像素坐标</span></span>
|
||
<span class="line"><span style="color:#B392F0"> std</span><span style="color:#E1E4E8">::cout </span><span style="color:#F97583"><<</span><span style="color:#E1E4E8"> (</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(x) </span><span style="color:#F97583"><<</span><span style="color:#9ECBFF"> ", "</span><span style="color:#F97583"> <<</span><span style="color:#E1E4E8"> (</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(y)</span><span style="color:#F97583"><<</span><span style="color:#9ECBFF">"</span><span style="color:#79B8FF">\n</span><span style="color:#9ECBFF">"</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glVertex2i</span><span style="color:#E1E4E8">((</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(x), (</span><span style="color:#F97583">int</span><span style="color:#E1E4E8">)</span><span style="color:#B392F0">round</span><span style="color:#E1E4E8">(y));</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glEnd</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 显示回调函数</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> display</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawDDALine</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">20</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glFlush</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 初始化 OpenGL 设置</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 设置背景颜色为白色</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glClearColor</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">1.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 设置投影矩阵为 2D 正交投影</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glMatrixMode</span><span style="color:#E1E4E8">(GL_PROJECTION);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glLoadIdentity</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 定义可视区域，左下角(0,0)，右上角(WIDTH, HEIGHT)</span></span>
|
||
<span class="line"><span style="color:#B392F0"> gluOrtho2D</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0.0</span><span style="color:#E1E4E8">, WIDTH, </span><span style="color:#79B8FF">0.0</span><span style="color:#E1E4E8">, HEIGHT);</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583">int</span><span style="color:#B392F0"> main</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> argc</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">char**</span><span style="color:#FFAB70"> argv</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 初始化 GLUT</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInit</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">&</span><span style="color:#E1E4E8">argc, argv);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 设置显示模式：单缓冲、RGB 颜色模式</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitDisplayMode</span><span style="color:#E1E4E8">(GLUT_SINGLE </span><span style="color:#F97583">|</span><span style="color:#E1E4E8"> GLUT_RGB);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 设置窗口大小和位置</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitWindowSize</span><span style="color:#E1E4E8">(WIDTH, HEIGHT);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitWindowPosition</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">100</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">100</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 创建窗口</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutCreateWindow</span><span style="color:#E1E4E8">(</span><span style="color:#9ECBFF">"DDA算法"</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 注册回调函数</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutDisplayFunc</span><span style="color:#E1E4E8">(display);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 初始化设置</span></span>
|
||
<span class="line"><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 进入主循环</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutMainLoop</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> return</span><span style="color:#79B8FF"> 0</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span></code></pre>
|
||
<h1 id="中点画线算法">中点画线算法</h1>
|
||
<h2 id="算法简介-1">算法简介</h2>
|
||
<p>只考虑当直线的斜率<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="6.585ex" height="2.26ex" role="img" focusable="false" viewBox="0 -749.5 2910.6 999" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-9-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-9-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-9-TEX-N-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path id="MJX-9-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-9-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D458" xlink:href="#MJX-9-TEX-I-1D458"></use></g><g data-mml-node="mo" transform="translate(799,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-9-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1354.8,0)"><use data-c="3C" xlink:href="#MJX-9-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(2410.6,0)"><use data-c="31" xlink:href="#MJX-9-TEX-N-31"></use></g></g></g></svg></mjx-container>时的情况，假设现在有一条直线<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="13.534ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5982.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-10-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-10-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-10-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-10-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-10-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-10-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-10-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-10-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-10-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-10-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-10-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-10-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-10-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="2C" xlink:href="#MJX-10-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3213.4,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-10-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-10-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(4222,0)"><use data-c="2C" xlink:href="#MJX-10-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(4666.7,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-10-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-10-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(5593.2,0)"><use data-c="29" xlink:href="#MJX-10-TEX-N-29"></use></g></g></g></svg></mjx-container>,那么第一个点一定是<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.144ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3157.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-11-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-11-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-11-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-11-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-11-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-11-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-11-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-11-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-11-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-11-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-11-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-11-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="29" xlink:href="#MJX-11-TEX-N-29"></use></g></g></g></svg></mjx-container>无疑，下一个点的<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-12-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-12-TEX-I-1D465"></use></g></g></g></svg></mjx-container>坐标为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="6.179ex" height="1.846ex" role="img" focusable="false" viewBox="0 -666 2731 816" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-13-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-13-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-13-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-13-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-13-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2B" xlink:href="#MJX-13-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2231,0)"><use data-c="31" xlink:href="#MJX-13-TEX-N-31"></use></g></g></g></svg></mjx-container>,<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-14-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-14-TEX-I-1D466"></use></g></g></g></svg></mjx-container>坐标要么为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.24ex" height="1.971ex" role="img" focusable="false" viewBox="0 -666 990 871" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-15-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-15-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-15-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(490,0)"><use data-c="31" xlink:href="#MJX-15-TEX-N-31"></use></g></g></g></svg></mjx-container>要么为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="6.137ex" height="1.971ex" role="img" focusable="false" viewBox="0 -666 2712.4 871" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-16-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-16-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-16-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-16-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(490,0)"><use data-c="31" xlink:href="#MJX-16-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1212.2,0)"><use data-c="2B" xlink:href="#MJX-16-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2212.4,0)"><use data-c="31" xlink:href="#MJX-16-TEX-N-31"></use></g></g></g></svg></mjx-container>。关键在于每次取下一个点时，是取前一个的<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.24ex" height="1.971ex" role="img" focusable="false" viewBox="0 -666 990 871" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-17-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-17-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-17-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(490,0)"><use data-c="31" xlink:href="#MJX-17-TEX-N-31"></use></g></g></g></svg></mjx-container>呢，还是<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="6.137ex" height="1.971ex" role="img" focusable="false" viewBox="0 -666 2712.4 871" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-18-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-18-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-18-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-18-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(490,0)"><use data-c="31" xlink:href="#MJX-18-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1212.2,0)"><use data-c="2B" xlink:href="#MJX-18-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2212.4,0)"><use data-c="31" xlink:href="#MJX-18-TEX-N-31"></use></g></g></g></svg></mjx-container>,这时一定是取直线上点最靠近的那个了，而判断取哪个点就用到了中点，我们将中点代入直线中 <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="55.003ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 24311.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-19-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-19-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-19-TEX-I-1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path id="MJX-19-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-19-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-19-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-19-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-19-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-19-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-19-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-19-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-19-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-19-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-19-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-19-TEX-N-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-19-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path id="MJX-19-TEX-I-1D450" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-19-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><use data-c="3D" xlink:href="#MJX-19-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1853.6,0)"><use data-c="1D439" xlink:href="#MJX-19-TEX-I-1D439"></use></g><g data-mml-node="mo" transform="translate(2602.6,0)"><use data-c="28" xlink:href="#MJX-19-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(2991.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-19-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(4222.3,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(5222.6,0)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5722.6,0)"><use data-c="2C" xlink:href="#MJX-19-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(6167.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-19-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(7316,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(8316.2,0)"><use data-c="30" xlink:href="#MJX-19-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-19-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-19-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mo" transform="translate(9594.2,0)"><use data-c="29" xlink:href="#MJX-19-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(10261,0)"><use data-c="3D" xlink:href="#MJX-19-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(11316.8,0)"><use data-c="1D44E" xlink:href="#MJX-19-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(12068,0)"><use data-c="22C5" xlink:href="#MJX-19-TEX-N-22C5"></use></g><g data-mml-node="mo" transform="translate(12568.2,0)"><use data-c="28" xlink:href="#MJX-19-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(12957.2,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-19-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(14188,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(15188.2,0)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(15688.2,0)"><use data-c="29" xlink:href="#MJX-19-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(16299.4,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(17299.7,0)"><use data-c="1D44F" xlink:href="#MJX-19-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(17950.9,0)"><use data-c="22C5" xlink:href="#MJX-19-TEX-N-22C5"></use></g><g data-mml-node="mo" transform="translate(18451.1,0)"><use data-c="28" xlink:href="#MJX-19-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(18840.1,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-19-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-19-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(19988.9,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(20989.1,0)"><use data-c="30" xlink:href="#MJX-19-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-19-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-19-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mo" transform="translate(22267.1,0)"><use data-c="29" xlink:href="#MJX-19-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(22878.3,0)"><use data-c="2B" xlink:href="#MJX-19-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(23878.5,0)"><use data-c="1D450" xlink:href="#MJX-19-TEX-I-1D450"></use></g></g></g></svg></mjx-container>。</p>
|
||
<ol>
|
||
<li>如果直线<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.085ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 3131.6 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-20-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-20-TEX-N-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path id="MJX-20-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-20-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-20-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><g data-mml-node="text"><use data-c="3E" xlink:href="#MJX-20-TEX-N-3E"></use></g><g data-mml-node="text" transform="translate(778,0)"><use data-c="3D" xlink:href="#MJX-20-TEX-N-3D"></use></g></g><g data-mml-node="mn" transform="translate(2631.6,0)"><use data-c="30" xlink:href="#MJX-20-TEX-N-30"></use></g></g></g></svg></mjx-container>，则取下边的点也就是<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="11.041ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4880.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-21-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-21-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-21-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-21-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-21-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-21-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-21-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-21-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-21-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-21-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1619.8,0)"><use data-c="2B" xlink:href="#MJX-21-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2620,0)"><use data-c="31" xlink:href="#MJX-21-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(3120,0)"><use data-c="2C" xlink:href="#MJX-21-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3564.7,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-21-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-21-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(4491.2,0)"><use data-c="29" xlink:href="#MJX-21-TEX-N-29"></use></g></g></g></svg></mjx-container>。</li>
|
||
<li>如果直线<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.09ex;" xmlns="http://www.w3.org/2000/svg" width="5.325ex" height="1.661ex" role="img" focusable="false" viewBox="0 -694 2353.6 734" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-22-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-22-TEX-N-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path id="MJX-22-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-22-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><use data-c="3C" xlink:href="#MJX-22-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(1853.6,0)"><use data-c="30" xlink:href="#MJX-22-TEX-N-30"></use></g></g></g></svg></mjx-container>,则取上边的点也就是<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="14.938ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6602.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-23-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-23-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-23-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-23-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-23-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-23-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-23-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-23-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-23-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-23-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1619.8,0)"><use data-c="2B" xlink:href="#MJX-23-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2620,0)"><use data-c="31" xlink:href="#MJX-23-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(3120,0)"><use data-c="2C" xlink:href="#MJX-23-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3564.7,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-23-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-23-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(4713.4,0)"><use data-c="2B" xlink:href="#MJX-23-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(5713.7,0)"><use data-c="31" xlink:href="#MJX-23-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(6213.7,0)"><use data-c="29" xlink:href="#MJX-23-TEX-N-29"></use></g></g></g></svg></mjx-container>。</li>
|
||
</ol>
|
||
<p>它的实际过程就是这样每次根据前边的点判断下一个点在哪，然后进行打亮，但这样每次判断的时候都得代入直线方程计算太麻烦了，我们将这俩种情况分别代入直线方程中可以找出规律：</p>
|
||
<ol>
|
||
<li>当直线<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.085ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 3131.6 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-24-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-24-TEX-N-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path id="MJX-24-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-24-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-24-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><g data-mml-node="text"><use data-c="3E" xlink:href="#MJX-24-TEX-N-3E"></use></g><g data-mml-node="text" transform="translate(778,0)"><use data-c="3D" xlink:href="#MJX-24-TEX-N-3D"></use></g></g><g data-mml-node="mn" transform="translate(2631.6,0)"><use data-c="30" xlink:href="#MJX-24-TEX-N-30"></use></g></g></g></svg></mjx-container>时，经过化解得<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="10.32ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 4561.6 844" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-25-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-25-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-25-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-25-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-25-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-25-TEX-I-1D451"></use></g><g data-mml-node="mn" transform="translate(553,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-25-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1234.3,0)"><use data-c="3D" xlink:href="#MJX-25-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2290.1,0)"><use data-c="1D451" xlink:href="#MJX-25-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(3032.3,0)"><use data-c="2B" xlink:href="#MJX-25-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(4032.6,0)"><use data-c="1D44E" xlink:href="#MJX-25-TEX-I-1D44E"></use></g></g></g></svg></mjx-container>;</li>
|
||
<li>当直线<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.09ex;" xmlns="http://www.w3.org/2000/svg" width="5.325ex" height="1.661ex" role="img" focusable="false" viewBox="0 -694 2353.6 734" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-26-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-26-TEX-N-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path id="MJX-26-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-26-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(797.8,0)"><use data-c="3C" xlink:href="#MJX-26-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(1853.6,0)"><use data-c="30" xlink:href="#MJX-26-TEX-N-30"></use></g></g></g></svg></mjx-container>时，经过化解得<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="14.057ex" height="1.91ex" role="img" focusable="false" viewBox="0 -694 6213 844" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-27-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-27-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-27-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-27-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-27-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-27-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-27-TEX-I-1D451"></use></g><g data-mml-node="mn" transform="translate(553,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-27-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1234.3,0)"><use data-c="3D" xlink:href="#MJX-27-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2290.1,0)"><use data-c="1D451" xlink:href="#MJX-27-TEX-I-1D451"></use></g><g data-mml-node="mo" transform="translate(3032.3,0)"><use data-c="2B" xlink:href="#MJX-27-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(4032.6,0)"><use data-c="1D44E" xlink:href="#MJX-27-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(4783.8,0)"><use data-c="2B" xlink:href="#MJX-27-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(5784,0)"><use data-c="1D44F" xlink:href="#MJX-27-TEX-I-1D44F"></use></g></g></g></svg></mjx-container>;</li>
|
||
<li>初始值<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="13.006ex" height="1.945ex" role="img" focusable="false" viewBox="0 -694 5748.6 859.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-28-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-28-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-28-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-28-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-28-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-28-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-28-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path><path id="MJX-28-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-28-TEX-I-1D451"></use></g><g data-mml-node="mn" transform="translate(553,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-28-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1234.3,0)"><use data-c="3D" xlink:href="#MJX-28-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2290.1,0)"><use data-c="1D44E" xlink:href="#MJX-28-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(3041.3,0)"><use data-c="2B" xlink:href="#MJX-28-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(4041.6,0)"><use data-c="30" xlink:href="#MJX-28-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-28-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-28-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mi" transform="translate(5319.6,0)"><use data-c="1D44F" xlink:href="#MJX-28-TEX-I-1D44F"></use></g></g></g></svg></mjx-container>。</li>
|
||
</ol>
|
||
<p>也就是说每次的增量要么为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-29-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-29-TEX-I-1D44E"></use></g></g></g></svg></mjx-container>,要么为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="4.933ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2180.4 776" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-30-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path id="MJX-30-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-30-TEX-I-1D44F" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-30-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJX-30-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1751.4,0)"><use data-c="1D44F" xlink:href="#MJX-30-TEX-I-1D44F"></use></g></g></g></svg></mjx-container>,那么这样判断的时候就简单多了，因为我们每次只是判断它的正负。所以给等式同时乘2，将其中浮点数0.5化为整数，这样硬件操作时无疑更快了。</p>
|
||
<h2 id="算法实现-1">算法实现</h2>
|
||
<pre class="astro-code github-dark" style="background-color:#24292e;color:#e1e4e8; overflow-x: auto; white-space: pre-wrap; word-wrap: break-word;" tabindex="0" data-language="cpp"><code><span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <GL/freeglut.h></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <iostream></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <cmath></span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 中点画线算法函数 (仅针对斜率 0 <= k <= 1 的情况进行演示，其它象限需类比处理)</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> drawLineMidpoint</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x0</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y0</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x1</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y1</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> a </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> y0 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> y1;</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> b </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> x1 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> x0;</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> d </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 2</span><span style="color:#F97583"> *</span><span style="color:#E1E4E8"> a </span><span style="color:#F97583">+</span><span style="color:#E1E4E8"> b;</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> d1 </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 2</span><span style="color:#F97583"> *</span><span style="color:#E1E4E8"> a;</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> d2 </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 2</span><span style="color:#F97583"> *</span><span style="color:#E1E4E8"> (a </span><span style="color:#F97583">+</span><span style="color:#E1E4E8"> b);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> x </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> x0, y </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> y0;</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#B392F0"> glBegin</span><span style="color:#E1E4E8">(GL_POINTS);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glVertex2i</span><span style="color:#E1E4E8">(x, y);</span><span style="color:#6A737D"> // 画起点</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> while</span><span style="color:#E1E4E8"> (x </span><span style="color:#F97583"><</span><span style="color:#E1E4E8"> x1) {</span></span>
|
||
<span class="line"><span style="color:#F97583"> if</span><span style="color:#E1E4E8"> (d </span><span style="color:#F97583"><</span><span style="color:#79B8FF"> 0</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> x</span><span style="color:#F97583">++</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> y</span><span style="color:#F97583">++</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> d </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> d2;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> } </span><span style="color:#F97583">else</span><span style="color:#E1E4E8"> {</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> x</span><span style="color:#F97583">++</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> d </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> d1;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glVertex2i</span><span style="color:#E1E4E8">(x, y);</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glEnd</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 渲染回调函数</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> display</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glClear</span><span style="color:#E1E4E8">(GL_COLOR_BUFFER_BIT);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glColor3f</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">1.0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 设置画笔颜色为白色</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 调用算法画一条直线 (x0, y0) 到 (x1, y1)</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 注意：此处的坐标对应屏幕像素坐标</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawLineMidpoint</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">450</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">300</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#B392F0"> glFlush</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 初始化设置</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glClearColor</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0.0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 背景设为黑色</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glMatrixMode</span><span style="color:#E1E4E8">(GL_PROJECTION);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glLoadIdentity</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 设置正交投影矩阵，使坐标系与窗口像素对应</span></span>
|
||
<span class="line"><span style="color:#B392F0"> gluOrtho2D</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">); </span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583">int</span><span style="color:#B392F0"> main</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> argc</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">char**</span><span style="color:#FFAB70"> argv</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInit</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">&</span><span style="color:#E1E4E8">argc, argv);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitDisplayMode</span><span style="color:#E1E4E8">(GLUT_SINGLE </span><span style="color:#F97583">|</span><span style="color:#E1E4E8"> GLUT_RGB);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitWindowSize</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitWindowPosition</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">100</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">100</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutCreateWindow</span><span style="color:#E1E4E8">(</span><span style="color:#9ECBFF">"Midpoint Line Algorithm - FreeGLUT"</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutDisplayFunc</span><span style="color:#E1E4E8">(display);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutMainLoop</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> return</span><span style="color:#79B8FF"> 0</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span></code></pre>
|
||
<h1 id="bresenham算法">Bresenham算法</h1>
|
||
<h2 id="算法简介-2">算法简介</h2>
|
||
<p>假设我们需要由<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.144ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3157.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-31-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-31-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-31-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-31-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-31-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-31-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-31-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-31-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-31-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-31-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-31-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-31-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="29" xlink:href="#MJX-31-TEX-N-29"></use></g></g></g></svg></mjx-container>这一点，绘画一直线至右下角的另一点<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.144ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3157.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-32-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-32-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-32-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-32-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-32-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-32-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-32-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-32-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-32-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-32-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-32-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-32-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="29" xlink:href="#MJX-32-TEX-N-29"></use></g></g></g></svg></mjx-container>，x,y分别代表其水平及垂直坐标，并且<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="17.305ex" height="1.783ex" role="img" focusable="false" viewBox="0 -583 7648.7 788" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-33-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-33-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-33-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-33-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-33-TEX-N-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path id="MJX-33-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-33-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-33-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2212" xlink:href="#MJX-33-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2231,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-33-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-33-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(3517.3,0)"><use data-c="3E" xlink:href="#MJX-33-TEX-N-3E"></use></g><g data-mml-node="msub" transform="translate(4573.1,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-33-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-33-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(5721.9,0)"><use data-c="2212" xlink:href="#MJX-33-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(6722.1,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-33-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-33-TEX-N-30"></use></g></g></g></g></svg></mjx-container>。在此我们使用电脑系统常用的坐标系，即<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-34-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-34-TEX-I-1D465"></use></g></g></g></svg></mjx-container>坐标值沿<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-35-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-35-TEX-I-1D465"></use></g></g></g></svg></mjx-container>轴向右增长，<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-36-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-36-TEX-I-1D466"></use></g></g></g></svg></mjx-container>坐标值沿<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-37-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-37-TEX-I-1D466"></use></g></g></g></svg></mjx-container>轴向下增长。</p>
|
||
<p>因此x及y之值分别向右及向下增加，而两点之水平距离为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="7.329ex" height="1.694ex" role="img" focusable="false" viewBox="0 -583 3239.6 748.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-38-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-38-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-38-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-38-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-38-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-38-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2212" xlink:href="#MJX-38-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2231,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-38-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-38-TEX-N-30"></use></g></g></g></g></g></svg></mjx-container>且垂直距离为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="6.958ex" height="1.783ex" role="img" focusable="false" viewBox="0 -583 3075.6 788" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-39-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-39-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-39-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-39-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-39-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-39-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(1148.8,0)"><use data-c="2212" xlink:href="#MJX-39-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2149,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-39-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-39-TEX-N-30"></use></g></g></g></g></g></svg></mjx-container>。由此得之，该线的斜率必定介乎于<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-40-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-40-TEX-N-30"></use></g></g></g></svg></mjx-container>至<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-41-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-41-TEX-N-31"></use></g></g></g></svg></mjx-container>之间。而此算法之目的，就是找出在<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.375ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.375ex" role="img" focusable="false" viewBox="0 -442 1008.6 607.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-42-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-42-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-42-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-42-TEX-N-30"></use></g></g></g></g></g></svg></mjx-container>与<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.339ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 1008.6 592" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-43-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-43-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-43-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-43-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container>之间，第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-44-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-44-TEX-I-1D465"></use></g></g></g></svg></mjx-container>行相对应的第<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-45-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-45-TEX-I-1D466"></use></g></g></g></svg></mjx-container>列，从而得出一像素点，使得该像素点的位置最接近原本的线。</p>
|
||
<p>对于由<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.144ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3157.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-46-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-46-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-46-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-46-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-46-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-46-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-46-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-46-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-46-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-46-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-46-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-46-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="29" xlink:href="#MJX-46-TEX-N-29"></use></g></g></g></svg></mjx-container>及<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.144ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3157.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-47-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-47-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-47-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-47-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-47-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-47-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-47-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-47-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-47-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="2C" xlink:href="#MJX-47-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1842.2,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-47-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-47-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(2768.8,0)"><use data-c="29" xlink:href="#MJX-47-TEX-N-29"></use></g></g></g></svg></mjx-container>两点所组成之直线，公式如下：
|
||
<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -1.045ex;" xmlns="http://www.w3.org/2000/svg" width="22.557ex" height="3.075ex" role="img" focusable="false" viewBox="0 -897.2 9970 1359.3" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-48-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-48-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-48-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-48-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-48-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-48-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-48-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-48-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-48-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(712.2,0)"><use data-c="2212" xlink:href="#MJX-48-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1712.4,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-48-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-48-TEX-N-30"></use></g></g></g><g data-mml-node="mo" transform="translate(2916.8,0)"><use data-c="3D" xlink:href="#MJX-48-TEX-N-3D"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(3972.6,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(278,485) scale(0.707)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-48-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-48-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(926.6,0)"><use data-c="2212" xlink:href="#MJX-48-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1704.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-48-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-48-TEX-N-30"></use></g></g></g></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-48-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-48-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><use data-c="2212" xlink:href="#MJX-48-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1786.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-48-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-48-TEX-N-30"></use></g></g></g></g><rect width="2176.4" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(6389,0)"><use data-c="28" xlink:href="#MJX-48-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(6778,0)"><use data-c="1D465" xlink:href="#MJX-48-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(7572.2,0)"><use data-c="2212" xlink:href="#MJX-48-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(8572.4,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-48-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-48-TEX-N-30"></use></g></g></g><g data-mml-node="mo" transform="translate(9581,0)"><use data-c="29" xlink:href="#MJX-48-TEX-N-29"></use></g></g></g></svg></mjx-container>
|
||
因此，对于每一点的x，其y的值是
|
||
<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -1.045ex;" xmlns="http://www.w3.org/2000/svg" width="18.431ex" height="3.075ex" role="img" focusable="false" viewBox="0 -897.2 8146.4 1359.3" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-49-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-49-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-49-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-49-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-49-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-49-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-49-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-49-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(278,485) scale(0.707)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-49-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-49-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(926.6,0)"><use data-c="2212" xlink:href="#MJX-49-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1704.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-49-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-49-TEX-N-30"></use></g></g></g></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-49-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-49-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><use data-c="2212" xlink:href="#MJX-49-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1786.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-49-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-49-TEX-N-30"></use></g></g></g></g><rect width="2176.4" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(2416.4,0)"><use data-c="28" xlink:href="#MJX-49-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2805.4,0)"><use data-c="1D465" xlink:href="#MJX-49-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(3599.7,0)"><use data-c="2212" xlink:href="#MJX-49-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(4599.9,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-49-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-49-TEX-N-30"></use></g></g></g><g data-mml-node="mo" transform="translate(5608.4,0)"><use data-c="29" xlink:href="#MJX-49-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6219.7,0)"><use data-c="2B" xlink:href="#MJX-49-TEX-N-2B"></use></g><g data-mml-node="msub" transform="translate(7219.9,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-49-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-49-TEX-N-30"></use></g></g></g></g></g></svg></mjx-container>
|
||
因为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-50-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-50-TEX-I-1D465"></use></g></g></g></svg></mjx-container>及<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-51-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-51-TEX-I-1D466"></use></g></g></g></svg></mjx-container>皆为整数，但并非每一点<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-52-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-52-TEX-I-1D465"></use></g></g></g></svg></mjx-container>所对应的<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-53-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-53-TEX-I-1D466"></use></g></g></g></svg></mjx-container>皆为整数，故此没有必要去计算每一点x所对应之<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-54-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-54-TEX-I-1D466"></use></g></g></g></svg></mjx-container>值。反之由于此线之斜率介乎于<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-55-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-55-TEX-N-31"></use></g></g></g></svg></mjx-container>至<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-56-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-56-TEX-N-30"></use></g></g></g></svg></mjx-container>之间，故此我们只需要找出当<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-57-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-57-TEX-I-1D465"></use></g></g></g></svg></mjx-container>到达那一个数值时，会使<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-58-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-58-TEX-I-1D466"></use></g></g></g></svg></mjx-container>上升<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-59-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-59-TEX-N-31"></use></g></g></g></svg></mjx-container>，若<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-60-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-60-TEX-I-1D465"></use></g></g></g></svg></mjx-container>尚未到此值，则<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-61-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-61-TEX-I-1D466"></use></g></g></g></svg></mjx-container>不变。至于如何找出相关的<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-62-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-62-TEX-I-1D465"></use></g></g></g></svg></mjx-container>值，则需依靠斜率。斜率之计算方法为<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="23.943ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10582.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-63-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-63-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-63-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-63-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-63-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-63-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-63-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-63-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-63-TEX-N-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path id="MJX-63-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-63-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(1155.8,0)"><use data-c="3D" xlink:href="#MJX-63-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2211.6,0)"><use data-c="28" xlink:href="#MJX-63-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(2600.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-63-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-63-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(3749.3,0)"><use data-c="2212" xlink:href="#MJX-63-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(4749.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-63-TEX-I-1D466"></use></g><g data-mml-node="TeXAtom" transform="translate(523,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-63-TEX-N-30"></use></g></g></g><g data-mml-node="mo" transform="translate(5676.1,0)"><use data-c="29" xlink:href="#MJX-63-TEX-N-29"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6065.1,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-63-TEX-N-2F"></use></g></g><g data-mml-node="mo" transform="translate(6565.1,0)"><use data-c="28" xlink:href="#MJX-63-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(6954.1,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-63-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-63-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(8184.9,0)"><use data-c="2212" xlink:href="#MJX-63-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(9185.1,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-63-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-63-TEX-N-30"></use></g></g></g><g data-mml-node="mo" transform="translate(10193.7,0)"><use data-c="29" xlink:href="#MJX-63-TEX-N-29"></use></g></g></g></svg></mjx-container>。由于此值不变，故可于运算前预先计算，减少运算次数。</p>
|
||
<p>要实行此算法，我们需计算每一像素点与该线之间的误差。于上述例子中，误差应为每一点<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-64-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-64-TEX-I-1D465"></use></g></g></g></svg></mjx-container>中，其相对的像素点之<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-65-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-65-TEX-I-1D466"></use></g></g></g></svg></mjx-container>值与该线实际之<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-66-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-66-TEX-I-1D466"></use></g></g></g></svg></mjx-container>值的差距。每当<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-67-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-67-TEX-I-1D466"></use></g></g></g></svg></mjx-container>的值增加<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-68-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-68-TEX-N-31"></use></g></g></g></svg></mjx-container>，误差的值就会增加<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.986ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 878 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-69-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-69-TEX-I-1D45A"></use></g></g></g></svg></mjx-container>。每当误差的值超出<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 1278 688" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-70-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-70-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path id="MJX-70-TEX-N-35" d="M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-70-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-70-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-70-TEX-N-35" transform="translate(778,0)"></use></g></g></g></svg></mjx-container>，线就会比较靠近下一个映像点，因此<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-71-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-71-TEX-I-1D466"></use></g></g></g></svg></mjx-container>的值便会加<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-72-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-72-TEX-N-31"></use></g></g></g></svg></mjx-container>，且误差减<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-73-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-73-TEX-N-31"></use></g></g></g></svg></mjx-container>。</p>
|
||
<h2 id="算法实现-2">算法实现</h2>
|
||
<pre class="astro-code github-dark" style="background-color:#24292e;color:#e1e4e8; overflow-x: auto; white-space: pre-wrap; word-wrap: break-word;" tabindex="0" data-language="cpp"><code><span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <GL/freeglut.h></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <iostream></span></span>
|
||
<span class="line"><span style="color:#F97583">#include</span><span style="color:#9ECBFF"> <cmath></span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 通用 Bresenham 画线算法</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> drawLineBresenham</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x0</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y0</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> x1</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">int</span><span style="color:#FFAB70"> y1</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> dx </span><span style="color:#F97583">=</span><span style="color:#B392F0"> abs</span><span style="color:#E1E4E8">(x1 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> x0);</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> dy </span><span style="color:#F97583">=</span><span style="color:#B392F0"> abs</span><span style="color:#E1E4E8">(y1 </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> y0);</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> sx </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> (x0 </span><span style="color:#F97583"><</span><span style="color:#E1E4E8"> x1) </span><span style="color:#F97583">?</span><span style="color:#79B8FF"> 1</span><span style="color:#F97583"> :</span><span style="color:#F97583"> -</span><span style="color:#79B8FF">1</span><span style="color:#E1E4E8">;</span><span style="color:#6A737D"> // X 方向步进</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> sy </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> (y0 </span><span style="color:#F97583"><</span><span style="color:#E1E4E8"> y1) </span><span style="color:#F97583">?</span><span style="color:#79B8FF"> 1</span><span style="color:#F97583"> :</span><span style="color:#F97583"> -</span><span style="color:#79B8FF">1</span><span style="color:#E1E4E8">;</span><span style="color:#6A737D"> // Y 方向步进</span></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> err </span><span style="color:#F97583">=</span><span style="color:#E1E4E8"> dx </span><span style="color:#F97583">-</span><span style="color:#E1E4E8"> dy;</span><span style="color:#6A737D"> // 初始误差项</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#B392F0"> glBegin</span><span style="color:#E1E4E8">(GL_POINTS);</span></span>
|
||
<span class="line"><span style="color:#F97583"> while</span><span style="color:#E1E4E8"> (</span><span style="color:#79B8FF">true</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glVertex2i</span><span style="color:#E1E4E8">(x0, y0);</span><span style="color:#6A737D"> // 绘制当前点</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> if</span><span style="color:#E1E4E8"> (x0 </span><span style="color:#F97583">==</span><span style="color:#E1E4E8"> x1 </span><span style="color:#F97583">&&</span><span style="color:#E1E4E8"> y0 </span><span style="color:#F97583">==</span><span style="color:#E1E4E8"> y1) </span><span style="color:#F97583">break</span><span style="color:#E1E4E8">;</span><span style="color:#6A737D"> // 到达终点</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583"> int</span><span style="color:#E1E4E8"> e2 </span><span style="color:#F97583">=</span><span style="color:#79B8FF"> 2</span><span style="color:#F97583"> *</span><span style="color:#E1E4E8"> err;</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 判断是否在 X 方向步进</span></span>
|
||
<span class="line"><span style="color:#F97583"> if</span><span style="color:#E1E4E8"> (e2 </span><span style="color:#F97583">></span><span style="color:#F97583"> -</span><span style="color:#E1E4E8">dy) {</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> err </span><span style="color:#F97583">-=</span><span style="color:#E1E4E8"> dy;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> x0 </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> sx;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#6A737D"> // 判断是否在 Y 方向步进</span></span>
|
||
<span class="line"><span style="color:#F97583"> if</span><span style="color:#E1E4E8"> (e2 </span><span style="color:#F97583"><</span><span style="color:#E1E4E8"> dx) {</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> err </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> dx;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> y0 </span><span style="color:#F97583">+=</span><span style="color:#E1E4E8"> sy;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#E1E4E8"> }</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glEnd</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D">// 渲染回调</span></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> display</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glClear</span><span style="color:#E1E4E8">(GL_COLOR_BUFFER_BIT);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glColor3f</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.8</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.4</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 设置一个好看的绿色（类似你图中的图标颜色）</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#6A737D"> // 绘制几条不同方向的线来测试算法的健壮性</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawLineBresenham</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">450</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">400</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 第一象限</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawLineBresenham</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">400</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">450</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 第四象限</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawLineBresenham</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">250</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">250</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">450</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 垂直线</span></span>
|
||
<span class="line"><span style="color:#B392F0"> drawLineBresenham</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">50</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">250</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">450</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">250</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 水平线</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#B392F0"> glFlush</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583">void</span><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">() {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glClearColor</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0.1</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.1</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0.1</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">1.0</span><span style="color:#F97583">f</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 深色背景</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glMatrixMode</span><span style="color:#E1E4E8">(GL_PROJECTION);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glLoadIdentity</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#B392F0"> gluOrtho2D</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">0</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">500</span><span style="color:#E1E4E8">);</span><span style="color:#6A737D"> // 建立 500x500 的直角坐标系</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span>
|
||
<span class="line"></span>
|
||
<span class="line"><span style="color:#F97583">int</span><span style="color:#B392F0"> main</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">int</span><span style="color:#FFAB70"> argc</span><span style="color:#E1E4E8">, </span><span style="color:#F97583">char**</span><span style="color:#FFAB70"> argv</span><span style="color:#E1E4E8">) {</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInit</span><span style="color:#E1E4E8">(</span><span style="color:#F97583">&</span><span style="color:#E1E4E8">argc, argv);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitDisplayMode</span><span style="color:#E1E4E8">(GLUT_SINGLE </span><span style="color:#F97583">|</span><span style="color:#E1E4E8"> GLUT_RGB);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutInitWindowSize</span><span style="color:#E1E4E8">(</span><span style="color:#79B8FF">600</span><span style="color:#E1E4E8">, </span><span style="color:#79B8FF">600</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutCreateWindow</span><span style="color:#E1E4E8">(</span><span style="color:#9ECBFF">"Bresenham Line Algorithm"</span><span style="color:#E1E4E8">);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> init</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutDisplayFunc</span><span style="color:#E1E4E8">(display);</span></span>
|
||
<span class="line"><span style="color:#B392F0"> glutMainLoop</span><span style="color:#E1E4E8">();</span></span>
|
||
<span class="line"><span style="color:#F97583"> return</span><span style="color:#79B8FF"> 0</span><span style="color:#E1E4E8">;</span></span>
|
||
<span class="line"><span style="color:#E1E4E8">}</span></span></code></pre><style>
|
||
mjx-container[jax="SVG"] {
|
||
direction: ltr;
|
||
}
|
||
|
||
mjx-container[jax="SVG"] > svg {
|
||
overflow: visible;
|
||
min-height: 1px;
|
||
min-width: 1px;
|
||
}
|
||
|
||
mjx-container[jax="SVG"] > svg a {
|
||
fill: blue;
|
||
stroke: blue;
|
||
}
|
||
|
||
mjx-container[jax="SVG"][display="true"] {
|
||
display: block;
|
||
text-align: center;
|
||
margin: 1em 0;
|
||
}
|
||
|
||
mjx-container[jax="SVG"][display="true"][width="full"] {
|
||
display: flex;
|
||
}
|
||
|
||
mjx-container[jax="SVG"][justify="left"] {
|
||
text-align: left;
|
||
}
|
||
|
||
mjx-container[jax="SVG"][justify="right"] {
|
||
text-align: right;
|
||
}
|
||
|
||
g[data-mml-node="merror"] > g {
|
||
fill: red;
|
||
stroke: red;
|
||
}
|
||
|
||
g[data-mml-node="merror"] > rect[data-background] {
|
||
fill: yellow;
|
||
stroke: none;
|
||
}
|
||
|
||
g[data-mml-node="mtable"] > line[data-line], svg[data-table] > g > line[data-line] {
|
||
stroke-width: 70px;
|
||
fill: none;
|
||
}
|
||
|
||
g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {
|
||
stroke-width: 70px;
|
||
fill: none;
|
||
}
|
||
|
||
g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {
|
||
stroke-dasharray: 140;
|
||
}
|
||
|
||
g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {
|
||
stroke-linecap: round;
|
||
stroke-dasharray: 0,140;
|
||
}
|
||
|
||
g[data-mml-node="mtable"] > g > svg {
|
||
overflow: visible;
|
||
}
|
||
|
||
[jax="SVG"] mjx-tool {
|
||
display: inline-block;
|
||
position: relative;
|
||
width: 0;
|
||
height: 0;
|
||
}
|
||
|
||
[jax="SVG"] mjx-tool > mjx-tip {
|
||
position: absolute;
|
||
top: 0;
|
||
left: 0;
|
||
}
|
||
|
||
mjx-tool > mjx-tip {
|
||
display: inline-block;
|
||
padding: .2em;
|
||
border: 1px solid #888;
|
||
font-size: 70%;
|
||
background-color: #F8F8F8;
|
||
color: black;
|
||
box-shadow: 2px 2px 5px #AAAAAA;
|
||
}
|
||
|
||
g[data-mml-node="maction"][data-toggle] {
|
||
cursor: pointer;
|
||
}
|
||
|
||
mjx-status {
|
||
display: block;
|
||
position: fixed;
|
||
left: 1em;
|
||
bottom: 1em;
|
||
min-width: 25%;
|
||
padding: .2em .4em;
|
||
border: 1px solid #888;
|
||
font-size: 90%;
|
||
background-color: #F8F8F8;
|
||
color: black;
|
||
}
|
||
|
||
foreignObject[data-mjx-xml] {
|
||
font-family: initial;
|
||
line-height: normal;
|
||
overflow: visible;
|
||
}
|
||
|
||
mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] {
|
||
stroke-width: 3;
|
||
}
|
||
</style> </div> <section class="post-copyright-card" aria-label="文章版权信息"> <div class="post-copyright-mark" aria-hidden="true"> <span>CC</span> </div> <div class="post-copyright-main"> <h2>OpenGL-直线的扫描转换</h2> <a class="post-copyright-link" href="/posts/7207243b/">/posts/7207243b/</a> <dl class="post-copyright-meta"> <div> <dt>作者</dt> <dd>biss</dd> </div> <div> <dt>发布于</dt> <dd><time datetime="2026-04-11T19:01:02.000Z">2026/04/12</time></dd> </div> <div> <dt>许可协议</dt> <dd><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank" rel="noreferrer">CC BY-NC-SA 4.0</a></dd> </div> </dl> </div> <div class="post-copyright-footer"> <div class="post-copyright-tags" aria-label="文章标签"> <a href="/tags/OpenGL/">OpenGL</a> </div> <div class="post-copyright-actions" aria-label="文章操作"> <button class="post-copyright-action is-primary post-share-button" type="button" data-share-title="OpenGL-直线的扫描转换" data-share-url="https://blog.biss.click/posts/7207243b/" title="分享文章"> <svg aria-hidden="true" viewBox="0 0 24 24"> <path d="M4 12v7a1 1 0 0 0 1 1h14a1 1 0 0 0 1-1v-7"></path> <path d="M12 16V4"></path> <path d="m7 9 5-5 5 5"></path> </svg> <span>分享</span> </button> <a class="post-copyright-action" href="/rss.xml" title="订阅 RSS"> <svg aria-hidden="true" viewBox="0 0 24 24"> <path d="M5 5a14 14 0 0 1 14 14"></path> <path d="M5 11a8 8 0 0 1 8 8"></path> <path d="M6 18h.01"></path> </svg> <span>订阅</span> </a> <a class="post-copyright-action" href="/about/" title="打赏作者"> <svg aria-hidden="true" viewBox="0 0 24 24"> <path d="M7 8h10"></path> <path d="M8 8V5h8v3"></path> <path d="M6 8h12l-1.5 11h-9L6 8Z"></path> <path d="M10 12h4"></path> </svg> <span>打赏</span> </a> <a class="post-copyright-icon" href="https://twitter.com/intent/tweet?url=https%3A%2F%2Fblog.biss.click%2Fposts%2F7207243b%2F&text=OpenGL-%E7%9B%B4%E7%BA%BF%E7%9A%84%E6%89%AB%E6%8F%8F%E8%BD%AC%E6%8D%A2" target="_blank" rel="noreferrer" title="分享到 X"> <span aria-hidden="true">x</span> </a> <a class="post-copyright-icon" href="https://service.weibo.com/share/share.php?url=https%3A%2F%2Fblog.biss.click%2Fposts%2F7207243b%2F&title=OpenGL-%E7%9B%B4%E7%BA%BF%E7%9A%84%E6%89%AB%E6%8F%8F%E8%BD%AC%E6%8D%A2" target="_blank" rel="noreferrer" title="分享到微博"> <span aria-hidden="true">微</span> </a> </div> </div> </section> <script>
|
||
(() => {
|
||
document.querySelectorAll('.post-share-button').forEach((button) => {
|
||
button.addEventListener('click', async () => {
|
||
const title = button.dataset.shareTitle || document.title;
|
||
const url = button.dataset.shareUrl || window.location.href;
|
||
const label = button.querySelector('span');
|
||
const originalText = label?.textContent || '分享';
|
||
|
||
try {
|
||
if (navigator.share) {
|
||
await navigator.share({ title, url });
|
||
return;
|
||
}
|
||
|
||
await navigator.clipboard.writeText(url);
|
||
if (label) label.textContent = '已复制';
|
||
} catch {
|
||
if (label) label.textContent = '复制失败';
|
||
} finally {
|
||
window.setTimeout(() => {
|
||
if (label) label.textContent = originalText;
|
||
}, 1800);
|
||
}
|
||
});
|
||
});
|
||
})();
|
||
</script> <section id="post-comment" class="twikoo-comments" aria-label="Comments" data-env-id="https://comment.biss.click" data-region data-path="/posts/7207243b/" data-script="https://cdn.jsdelivr.net/npm/twikoo@1.7.7/dist/twikoo.min.js"> <div class="twikoo-comments-head"> <span class="twikoo-comments-icon" aria-hidden="true"> <svg viewBox="0 0 24 24"> <path d="M21 12a8 8 0 0 1-8 8H7l-4 3v-5a8 8 0 1 1 18-6Z"></path> </svg> </span> <div> <h2>Comments</h2> <p>OpenGL-直线的扫描转换</p> </div> </div> <div id="twikoo-wrap" class="twikoo-wrap"></div> </section> <script>
|
||
(() => {
|
||
const root = document.getElementById('post-comment');
|
||
const wrap = document.getElementById('twikoo-wrap');
|
||
if (!root || !wrap || root.dataset.twikooLoaded === 'true') return;
|
||
|
||
const envId = root.dataset.envId;
|
||
const path = root.dataset.path || window.location.pathname;
|
||
const region = root.dataset.region || undefined;
|
||
if (!envId) return;
|
||
|
||
root.dataset.twikooLoaded = 'true';
|
||
|
||
const initTwikoo = () => {
|
||
if (!window.twikoo || typeof window.twikoo.init !== 'function') return;
|
||
|
||
window.twikoo.init({
|
||
el: wrap,
|
||
envId,
|
||
region,
|
||
path,
|
||
});
|
||
};
|
||
|
||
if (window.twikoo) {
|
||
initTwikoo();
|
||
return;
|
||
}
|
||
|
||
const script = document.createElement('script');
|
||
script.src = root.dataset.script;
|
||
script.defer = true;
|
||
script.onload = initTwikoo;
|
||
script.onerror = () => {
|
||
root.dataset.twikooLoaded = 'false';
|
||
};
|
||
document.head.appendChild(script);
|
||
})();
|
||
</script> </article> <aside class="article-sidebar" aria-label="文章侧边栏"> <section class="sidebar-card article-author-card"> <img class="profile-avatar" src="https://free.picui.cn/free/2025/08/10/689845496a283.png" alt="biss" loading="lazy"> <div class="profile-name">biss</div> <div class="article-author-meta"> <span>2026/04/12</span> <a href="/categories/tech/">技术</a> </div> </section> <section class="sidebar-card article-tags-card"> <h2> <span aria-hidden="true">#</span>
|
||
文章标签
|
||
<em>1</em> </h2> <div class="article-tag-list" aria-label="文章标签"> <a href="/tags/OpenGL/">#OpenGL</a> </div> </section> <div class="article-sticky-sidebar"> <section class="sidebar-card toc-card"> <h2> <span aria-hidden="true">☰</span>
|
||
文章目录
|
||
<em>9</em> </h2> <nav class="toc-list toc-tree" aria-label="文章目录" data-toc-tree> <div class="toc-group is-open" data-toc-group data-heading-id="dda数值微分线段算法"> <a class="toc-link toc-link-main" href="#dda数值微分线段算法" data-toc-link data-toc-top> <span class="toc-marker" aria-hidden="true"></span> <span>DDA数值微分线段算法</span> </a> <div class="toc-children" data-toc-children> <a class="toc-link toc-link-child toc-depth-2" href="#算法简介" data-toc-link> <span>算法简介</span> </a><a class="toc-link toc-link-child toc-depth-2" href="#算法实现" data-toc-link> <span>算法实现</span> </a> </div> </div><div class="toc-group" data-toc-group data-heading-id="中点画线算法"> <a class="toc-link toc-link-main" href="#中点画线算法" data-toc-link data-toc-top> <span class="toc-marker" aria-hidden="true"></span> <span>中点画线算法</span> </a> <div class="toc-children" data-toc-children> <a class="toc-link toc-link-child toc-depth-2" href="#算法简介-1" data-toc-link> <span>算法简介</span> </a><a class="toc-link toc-link-child toc-depth-2" href="#算法实现-1" data-toc-link> <span>算法实现</span> </a> </div> </div><div class="toc-group" data-toc-group data-heading-id="bresenham算法"> <a class="toc-link toc-link-main" href="#bresenham算法" data-toc-link data-toc-top> <span class="toc-marker" aria-hidden="true"></span> <span>Bresenham算法</span> </a> <div class="toc-children" data-toc-children> <a class="toc-link toc-link-child toc-depth-2" href="#算法简介-2" data-toc-link> <span>算法简介</span> </a><a class="toc-link toc-link-child toc-depth-2" href="#算法实现-2" data-toc-link> <span>算法实现</span> </a> </div> </div> </nav> </section> <section class="sidebar-card same-category-card"> <h2> <span aria-hidden="true">▣</span>
|
||
同类文章
|
||
<em>8</em> </h2> <div class="same-category-list"> <a href="/posts/3f8de22/"> <strong>GPT Image2 初试</strong> <time datetime="2026-04-25T15:00:00.000Z">2026/04/25</time> </a><a href="/posts/8712baaa/"> <strong>电子价签玩法</strong> <time datetime="2026-04-19T13:46:25.000Z">2026/04/19</time> </a><a href="/posts/9b2ef00a/"> <strong>OpenGL- Cohen-Sutherland 编码裁剪算法</strong> <time datetime="2026-04-15T19:34:23.000Z">2026/04/16</time> </a><a href="/posts/437a5198/"> <strong>OpenGL-基础程序</strong> <time datetime="2026-04-11T18:20:34.000Z">2026/04/12</time> </a><a href="/posts/c4477b0c/"> <strong>配置OpenGL环境</strong> <time datetime="2026-04-02T04:52:16.000Z">2026/04/02</time> </a><a href="/posts/b559997d/"> <strong>opencv应用-算术运算</strong> <time datetime="2026-03-17T13:13:08.000Z">2026/03/17</time> </a><a href="/posts/e8f95ead/"> <strong>opencv应用-基础操作</strong> <time datetime="2026-03-15T09:56:51.000Z">2026/03/15</time> </a><a href="/posts/6e3332b/"> <strong>配置opencv</strong> <time datetime="2026-03-14T11:48:36.000Z">2026/03/14</time> </a> </div> </section> <section class="sidebar-card site-info-card" data-site-info-card data-stats-endpoint="https://blogumami.biss.click"> <h2><i class="fa-solid fa-chart-line" aria-hidden="true"></i> 网站信息</h2> <dl class="site-info-list"> <div> <dt>文章数目</dt> <dd>44</dd> </div><div> <dt>本站总字数</dt> <dd>36,128</dd> </div><div> <dt>本站访客数</dt> <dd data-stat-key="visitors">--</dd> </div><div> <dt>本站访问量</dt> <dd data-stat-key="pageviews">--</dd> </div><div> <dt>最后更新时间</dt> <dd>2026/06/20</dd> </div> </dl> </section> <script>
|
||
(() => {
|
||
const cacheDuration = 10 * 60 * 1000;
|
||
const formatter = new Intl.NumberFormat('en-US');
|
||
const cards = [...document.querySelectorAll('[data-site-info-card]')];
|
||
const endpoint = cards.find((card) => card.dataset.statsEndpoint)?.dataset.statsEndpoint;
|
||
|
||
if (!endpoint) return;
|
||
|
||
const cacheKey = `site-info-stats:${endpoint}`;
|
||
|
||
const updateCards = (stats) => {
|
||
const visitors = Number(stats?.visitors);
|
||
const pageviews = Number(stats?.pageviews);
|
||
|
||
cards.forEach((card) => {
|
||
const visitorsEl = card.querySelector('[data-stat-key="visitors"]');
|
||
const pageviewsEl = card.querySelector('[data-stat-key="pageviews"]');
|
||
if (visitorsEl && Number.isFinite(visitors)) visitorsEl.textContent = formatter.format(visitors);
|
||
if (pageviewsEl && Number.isFinite(pageviews)) pageviewsEl.textContent = formatter.format(pageviews);
|
||
});
|
||
};
|
||
|
||
try {
|
||
const cached = JSON.parse(localStorage.getItem(cacheKey) || 'null');
|
||
if (cached?.time && Date.now() - cached.time < cacheDuration) {
|
||
updateCards(cached.data);
|
||
return;
|
||
}
|
||
} catch {
|
||
localStorage.removeItem(cacheKey);
|
||
}
|
||
|
||
fetch(endpoint, { headers: { Accept: 'application/json' } })
|
||
.then((response) => {
|
||
if (!response.ok) throw new Error(`Stats request failed: ${response.status}`);
|
||
return response.json();
|
||
})
|
||
.then((data) => {
|
||
localStorage.setItem(cacheKey, JSON.stringify({ time: Date.now(), data }));
|
||
updateCards(data);
|
||
})
|
||
.catch(() => {
|
||
});
|
||
})();
|
||
</script> </div> </aside> </section> </main> <footer class="site-footer" aria-label="站点页脚"> <div class="footer-inner"> <section class="footer-top" aria-label="站点信息"> <a class="footer-brand" href="/" aria-label="Bi's Blog 首页"> <img src="/images/Bi.ico" alt="" loading="lazy"> <span>Bi's Blog</span> </a> <p class="footer-slogan">好奇心转化为代码的空间。</p> </section> <section class="footer-contact" aria-label="联系方式"> <p>联系方式</p> <div class="footer-socials"> <a href="mailto:bishsh2006@gmail.com" aria-label="发送邮件" title="发送邮件"> <i class="fa-solid fa-envelope" aria-hidden="true"></i> </a><a href="/rss.xml" aria-label="订阅 RSS" title="订阅 RSS"> <i class="fa-solid fa-rss" aria-hidden="true"></i> </a><a href="https://github.com/bishshi" target="_blank" rel="noreferrer" aria-label="GitHub" title="GitHub"> <i class="fa-brands fa-github" aria-hidden="true"></i> </a> </div> <nav class="footer-contact-links" aria-label="页脚链接"> <a href="/archives/">发展历程</a> <span aria-hidden="true">|</span><a href="/privacy/">隐私政策</a> </nav> </section> <section class="footer-bottom" aria-label="版权信息"> <div> <p>© 2025 - 2026 Bi's Blog</p> <p>blog.biss.click | 内容采用 CC BY-NC-SA 4.0 协议</p> </div> <div class="footer-meta"> <p>本站已运行：<span id="site-runtime">计算中...</span></p> <p> <a href="https://fastly.com/" target="_blank" rel="noreferrer">Fastly</a> <span aria-hidden="true">|</span><a href="https://astro.build/" target="_blank" rel="noreferrer">ASTRO</a> </p> </div> </section> </div> </footer> <aside class="hitokoto-card is-collapsed" aria-label="一言" data-hitokoto-card data-astro-cid-4cgfuthx> <div class="hitokoto-card__head" data-astro-cid-4cgfuthx> <button class="hitokoto-card__toggle" type="button" aria-label="展开一言" aria-expanded="false" title="一言" data-hitokoto-toggle data-astro-cid-4cgfuthx> <i class="fa-solid fa-quote-left" aria-hidden="true" data-astro-cid-4cgfuthx></i> <span data-astro-cid-4cgfuthx>一言</span> </button> <div class="hitokoto-card__actions" data-astro-cid-4cgfuthx> <button class="hitokoto-card__refresh" type="button" aria-label="换一句" title="换一句" data-hitokoto-refresh data-astro-cid-4cgfuthx> <i class="fa-solid fa-rotate" aria-hidden="true" data-astro-cid-4cgfuthx></i> </button> <button class="hitokoto-card__collapse" type="button" aria-label="收起一言" title="收起" data-hitokoto-collapse data-astro-cid-4cgfuthx> <i class="fa-solid fa-chevron-down" aria-hidden="true" data-astro-cid-4cgfuthx></i> </button> </div> </div> <a class="hitokoto-card__text" href="https://hitokoto.cn/" target="_blank" rel="noreferrer" data-hitokoto-text data-astro-cid-4cgfuthx>
|
||
获取中...
|
||
</a> <p class="hitokoto-card__from" data-hitokoto-from data-astro-cid-4cgfuthx>来自 Hitokoto</p> </aside> <script type="module">(()=>{const o=document.querySelector("[data-hitokoto-card]"),c=o?.querySelector("[data-hitokoto-toggle]"),s=o?.querySelector("[data-hitokoto-text]"),l=o?.querySelector("[data-hitokoto-from]"),g=o?.querySelector("[data-hitokoto-refresh]"),k=o?.querySelector("[data-hitokoto-collapse]");if(!(o instanceof HTMLElement)||!(c instanceof HTMLButtonElement)||!(s instanceof HTMLAnchorElement)||!(l instanceof HTMLElement))return;const y="https://yiyan.biss.click/api?encode=json",u="hitokoto-card-cache",d="hitokoto-card-expanded",x=360*60*1e3,f={sentence:"生活明朗，万物可爱。",source:"本地缓存",url:"https://hitokoto.cn/"},S=["/","/index.html"].includes(window.location.pathname),w=window.matchMedia("(max-width: 720px)");function E(){try{const t=localStorage.getItem(d);if(t==="true")return!0;if(t==="false")return!1}catch{}return null}function L(t){try{localStorage.setItem(d,String(t))}catch{}}function r(t,{persist:e=!1}={}){o.classList.toggle("is-collapsed",!t),c.setAttribute("aria-expanded",String(t)),c.setAttribute("aria-label",t?"收起一言":"展开一言"),e&&L(t)}function h(){try{const t=localStorage.getItem(u);if(!t)return null;const e=JSON.parse(t);return!e||typeof e.expiresAt!="number"||e.expiresAt<=Date.now()?null:e.data}catch{return null}}function H(t){try{localStorage.setItem(u,JSON.stringify({data:t,expiresAt:Date.now()+x}))}catch{}}function i(t){s.textContent=t.sentence,s.href=t.url||"https://hitokoto.cn/",l.textContent=t.source?`出自 ${t.source}`:"来自 Hitokoto"}async function p({force:t=!1}={}){if(!t){const e=h();if(e){i(e);return}}o.classList.add("is-loading"),s.textContent="获取中...";try{const e=await fetch(y);if(!e.ok)throw new Error(`Hitokoto request failed: ${e.status}`);const n=await e.json(),a=n.hitokoto||n.content||n.sentence||n.text,q=typeof a=="string"&&a.trim()?a.trim():f.sentence,A=typeof n.from=="string"&&n.from.trim()?n.from.trim():"Hitokoto",M=n.uuid?`https://hitokoto.cn/?uuid=${encodeURIComponent(n.uuid)}`:"https://hitokoto.cn/",m={sentence:q,source:A,url:M};H(m),i(m)}catch(e){console.error(e),i(h()||f)}finally{o.classList.remove("is-loading")}}const b=E();r(b??(S&&!w.matches)),c.addEventListener("click",()=>{r(o.classList.contains("is-collapsed"),{persist:!0})}),g?.addEventListener("click",t=>{t.stopPropagation(),r(!0,{persist:!0}),p({force:!0})}),k?.addEventListener("click",t=>{t.stopPropagation(),r(!1,{persist:!0})}),o.addEventListener("click",t=>{o.classList.contains("is-collapsed")&&(t.target instanceof Element&&t.target.closest("a, button")||r(!0,{persist:!0}))}),p()})();</script> <div class="custom-context-menu" data-context-menu role="menu" aria-hidden="true"> <div class="context-menu-nav" aria-label="页面导航"> <button type="button" data-context-action="back" aria-label="后退上一页" title="后退上一页"> <i class="fa-solid fa-arrow-left" aria-hidden="true"></i> </button> <button type="button" data-context-action="forward" aria-label="前进下一页" title="前进下一页"> <i class="fa-solid fa-arrow-right" aria-hidden="true"></i> </button> <button type="button" data-context-action="reload" aria-label="刷新页面" title="刷新页面"> <i class="fa-solid fa-rotate-right" aria-hidden="true"></i> </button> <a href="/" aria-label="返回首页" title="返回首页"> <i class="fa-solid fa-house" aria-hidden="true"></i> </a> </div> <div class="context-menu-group" data-selection-group hidden> <button type="button" data-context-action="copy-selection" data-selection-item> <i class="fa-regular fa-copy" aria-hidden="true"></i> <span>复制选中文字</span> </button> <button type="button" data-context-action="search-selection" data-selection-item> <i class="fa-brands fa-microsoft" aria-hidden="true"></i> <span>用 Bing 搜索</span> </button> <button type="button" data-context-action="quote-selection" data-selection-item data-comments-item> <i class="fa-regular fa-comment-dots" aria-hidden="true"></i> <span>引用到评论区</span> </button> </div> <div class="context-menu-group" data-image-group hidden> <button type="button" data-context-action="copy-image"> <i class="fa-regular fa-image" aria-hidden="true"></i> <span>复制图片</span> </button> <button type="button" data-context-action="open-image"> <i class="fa-solid fa-up-right-from-square" aria-hidden="true"></i> <span>新页面打开图片</span> </button> <button type="button" data-context-action="copy-image-url"> <i class="fa-solid fa-link" aria-hidden="true"></i> <span>复制图片地址</span> </button> </div> <div class="context-menu-group"> <a href="/archives/"> <i class="fa-solid fa-box-archive" aria-hidden="true"></i> <span>时间线</span> </a> <a href="/categories/"> <i class="fa-solid fa-folder" aria-hidden="true"></i> <span>分类</span> </a> <a href="/tags/"> <i class="fa-solid fa-hashtag" aria-hidden="true"></i> <span>标签</span> </a> <button type="button" data-context-action="random-post"> <i class="fa-solid fa-shuffle" aria-hidden="true"></i> <span>随机文章</span> </button> <button type="button" data-context-action="comments" data-comments-item> <i class="fa-regular fa-comment-dots" aria-hidden="true"></i> <span>跳转评论区</span> </button> <button type="button" data-context-action="print"> <i class="fa-solid fa-print" aria-hidden="true"></i> <span>打印页面</span> </button> <a href="/cc/"> <i class="fa-solid fa-circle-info" aria-hidden="true"></i> <span>网站版权</span> </a> </div> </div> <div class="context-menu-native-hint" data-context-native-hint role="note" aria-label="Ctrl 右键提示"> <kbd>Ctrl</kbd> + 右键：浏览器原生菜单
|
||
</div> <script>
|
||
(() => {
|
||
const menu = document.querySelector('[data-context-menu]');
|
||
if (!menu) return;
|
||
|
||
const nativeHint = document.querySelector('[data-context-native-hint]');
|
||
const selectionItems = [...menu.querySelectorAll('[data-selection-item]')];
|
||
const selectionGroup = menu.querySelector('[data-selection-group]');
|
||
const commentsItems = [...menu.querySelectorAll('[data-comments-item]')];
|
||
const imageGroup = menu.querySelector('[data-image-group]');
|
||
const focusableSelector = 'a[href], button:not([disabled])';
|
||
let lastSelectionText = '';
|
||
let lastImage = null;
|
||
let nativeHintTimer = 0;
|
||
|
||
const isNativeContextTarget = (target) =>
|
||
target instanceof Element &&
|
||
Boolean(target.closest('input, textarea, select, [contenteditable="true"], [contenteditable=""]'));
|
||
|
||
const getCommentsTarget = () =>
|
||
document.querySelector('.twikoo-comments, #twikoo-wrap, #twikoo, .tk-comments-container');
|
||
|
||
const getSelectionText = () => String(window.getSelection?.() || '').trim();
|
||
|
||
const getImageTarget = (target) => {
|
||
if (!(target instanceof Element)) return null;
|
||
const image = target.closest('img');
|
||
return image instanceof HTMLImageElement ? image : null;
|
||
};
|
||
|
||
const getImageUrl = () => {
|
||
if (!lastImage) return '';
|
||
return lastImage.currentSrc || lastImage.src || lastImage.getAttribute('src') || '';
|
||
};
|
||
|
||
const copyText = async (text) => {
|
||
if (navigator.clipboard?.writeText) {
|
||
await navigator.clipboard.writeText(text);
|
||
return;
|
||
}
|
||
|
||
const textarea = document.createElement('textarea');
|
||
textarea.value = text;
|
||
textarea.setAttribute('readonly', '');
|
||
textarea.style.position = 'fixed';
|
||
textarea.style.inset = '0 auto auto -9999px';
|
||
document.body.appendChild(textarea);
|
||
textarea.select();
|
||
document.execCommand('copy');
|
||
textarea.remove();
|
||
};
|
||
|
||
const setMenuState = () => {
|
||
const hasSelection = lastSelectionText.length > 0;
|
||
const hasComments = Boolean(getCommentsTarget());
|
||
const hasImage = Boolean(getImageUrl());
|
||
|
||
selectionItems.forEach((item) => {
|
||
item.hidden = !hasSelection;
|
||
});
|
||
if (selectionGroup) selectionGroup.hidden = !hasSelection;
|
||
|
||
commentsItems.forEach((item) => {
|
||
item.hidden = !hasComments || (item.hasAttribute('data-selection-item') && !hasSelection);
|
||
});
|
||
|
||
if (imageGroup) imageGroup.hidden = !hasImage;
|
||
};
|
||
|
||
const closeMenu = () => {
|
||
menu.classList.remove('is-open');
|
||
menu.setAttribute('aria-hidden', 'true');
|
||
};
|
||
|
||
const openMenu = (x, y) => {
|
||
setMenuState();
|
||
menu.classList.add('is-open');
|
||
menu.setAttribute('aria-hidden', 'false');
|
||
|
||
const rect = menu.getBoundingClientRect();
|
||
const gap = 10;
|
||
const left = Math.min(Math.max(gap, x), window.innerWidth - rect.width - gap);
|
||
const top = Math.min(Math.max(gap, y), window.innerHeight - rect.height - gap);
|
||
|
||
menu.style.left = `${left}px`;
|
||
menu.style.top = `${top}px`;
|
||
};
|
||
|
||
const showNativeHint = () => {
|
||
if (!nativeHint) return;
|
||
window.clearTimeout(nativeHintTimer);
|
||
nativeHint.classList.add('is-visible');
|
||
nativeHintTimer = window.setTimeout(() => {
|
||
nativeHint.classList.remove('is-visible');
|
||
}, 2600);
|
||
};
|
||
|
||
const jumpToComments = () => {
|
||
const target = getCommentsTarget();
|
||
target?.scrollIntoView({ behavior: 'smooth', block: 'start' });
|
||
};
|
||
|
||
const quoteToComments = () => {
|
||
jumpToComments();
|
||
window.setTimeout(() => {
|
||
const textarea = document.querySelector('#twikoo textarea, .twikoo-comments textarea, .tk-input textarea');
|
||
if (!(textarea instanceof HTMLTextAreaElement) || !lastSelectionText) return;
|
||
|
||
const quote = lastSelectionText
|
||
.split(/\r?\n/)
|
||
.map((line) => `> ${line}`)
|
||
.join('\n');
|
||
const current = textarea.value.trim();
|
||
textarea.value = current ? `${current}\n\n${quote}\n` : `${quote}\n`;
|
||
textarea.dispatchEvent(new Event('input', { bubbles: true }));
|
||
textarea.focus();
|
||
}, 500);
|
||
};
|
||
|
||
const searchSelection = () => {
|
||
if (!lastSelectionText) return;
|
||
const url = `https://www.bing.com/search?q=${encodeURIComponent(lastSelectionText)}`;
|
||
window.open(url, '_blank', 'noopener,noreferrer');
|
||
};
|
||
|
||
const copyImage = async () => {
|
||
const imageUrl = getImageUrl();
|
||
if (!imageUrl) return;
|
||
|
||
try {
|
||
if (!navigator.clipboard?.write || typeof ClipboardItem === 'undefined') {
|
||
throw new Error('Image clipboard is not available.');
|
||
}
|
||
|
||
const response = await fetch(imageUrl, { mode: 'cors' });
|
||
const blob = await response.blob();
|
||
await navigator.clipboard.write([
|
||
new ClipboardItem({ [blob.type || 'image/png']: blob }),
|
||
]);
|
||
} catch {
|
||
await copyText(new URL(imageUrl, window.location.href).toString());
|
||
}
|
||
};
|
||
|
||
const openImage = () => {
|
||
const imageUrl = getImageUrl();
|
||
if (!imageUrl) return;
|
||
window.open(new URL(imageUrl, window.location.href).toString(), '_blank', 'noopener,noreferrer');
|
||
};
|
||
|
||
document.addEventListener('contextmenu', (event) => {
|
||
if (event.ctrlKey) {
|
||
closeMenu();
|
||
nativeHint?.classList.remove('is-visible');
|
||
return;
|
||
}
|
||
|
||
if (isNativeContextTarget(event.target)) return;
|
||
|
||
lastSelectionText = getSelectionText();
|
||
lastImage = getImageTarget(event.target);
|
||
event.preventDefault();
|
||
openMenu(event.clientX, event.clientY);
|
||
showNativeHint();
|
||
});
|
||
|
||
document.addEventListener('click', (event) => {
|
||
const actionTarget = event.target instanceof Element ? event.target.closest('[data-context-action]') : null;
|
||
if (!actionTarget || !menu.contains(actionTarget)) {
|
||
closeMenu();
|
||
return;
|
||
}
|
||
|
||
const action = actionTarget.dataset.contextAction;
|
||
closeMenu();
|
||
|
||
if (action === 'back') history.back();
|
||
if (action === 'forward') history.forward();
|
||
if (action === 'reload') location.reload();
|
||
if (action === 'copy-selection' && lastSelectionText) copyText(lastSelectionText);
|
||
if (action === 'search-selection') searchSelection();
|
||
if (action === 'quote-selection') quoteToComments();
|
||
if (action === 'copy-image') copyImage();
|
||
if (action === 'open-image') openImage();
|
||
if (action === 'copy-image-url') copyText(new URL(getImageUrl(), window.location.href).toString());
|
||
if (action === 'random-post' && typeof window.randomPost === 'function') window.randomPost(actionTarget);
|
||
if (action === 'comments') jumpToComments();
|
||
if (action === 'print') window.print();
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape') closeMenu();
|
||
if (event.key === 'Tab' && menu.classList.contains('is-open')) {
|
||
const items = [...menu.querySelectorAll(focusableSelector)].filter((item) => !item.closest('[hidden]'));
|
||
if (items.length === 0) return;
|
||
const first = items[0];
|
||
const last = items[items.length - 1];
|
||
|
||
if (event.shiftKey && document.activeElement === first) {
|
||
event.preventDefault();
|
||
last.focus();
|
||
} else if (!event.shiftKey && document.activeElement === last) {
|
||
event.preventDefault();
|
||
first.focus();
|
||
}
|
||
}
|
||
});
|
||
|
||
window.addEventListener('scroll', closeMenu, { passive: true });
|
||
window.addEventListener('resize', closeMenu);
|
||
})();
|
||
</script> <aside class="pwa-toast" data-pwa-toast hidden role="status" aria-live="polite"> <div class="pwa-toast-icon" aria-hidden="true"><i class="fa-solid fa-download"></i></div> <div class="pwa-toast-copy"> <strong data-pwa-title>PWA 提示</strong> <p data-pwa-message></p> </div> <div class="pwa-toast-actions"> <button class="pwa-toast-primary" type="button" data-pwa-primary>确定</button> <button class="pwa-toast-close" type="button" data-pwa-close aria-label="关闭提示">稍后</button> </div> </aside> <script src="https://cdn.jsdmirror.com/npm/instantsearch.js@4.56.0"></script> <script src="https://cdn.jsdmirror.com/npm/typesense-instantsearch-adapter@2.7.0/dist/typesense-instantsearch-adapter.min.js"></script> <script src="https://cdn.jsdmirror.com/npm/@fancyapps/ui@5.0/dist/fancybox/fancybox.umd.js"></script> <script src="/js/random.js"></script> <script src="/js/search/typesense-search.js"></script> <script src="/js/external-links.js"></script> <script src="/js/pwa.js"></script> <script>
|
||
(() => {
|
||
const nav = document.querySelector('.nav');
|
||
const toggle = document.querySelector('[data-nav-toggle]');
|
||
const menu = document.querySelector('[data-nav-menu]');
|
||
const backdrop = document.querySelector('[data-nav-backdrop]');
|
||
if (!nav || !toggle || !menu) return;
|
||
|
||
function setMenu(open) {
|
||
nav.classList.toggle('is-open', open);
|
||
toggle.setAttribute('aria-expanded', String(open));
|
||
toggle.setAttribute('aria-label', open ? '关闭菜单' : '打开菜单');
|
||
menu.setAttribute('aria-hidden', String(!open && window.innerWidth <= 720));
|
||
document.body.classList.toggle('has-nav-drawer', open);
|
||
}
|
||
|
||
toggle.addEventListener('click', () => {
|
||
setMenu(!nav.classList.contains('is-open'));
|
||
});
|
||
|
||
backdrop?.addEventListener('click', () => setMenu(false));
|
||
|
||
menu.addEventListener('click', (event) => {
|
||
if (event.target instanceof Element && event.target.closest('a')) setMenu(false);
|
||
});
|
||
|
||
document.addEventListener('click', (event) => {
|
||
if (!nav.classList.contains('is-open')) return;
|
||
if (event.target instanceof Node && nav.contains(event.target)) return;
|
||
setMenu(false);
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape' && nav.classList.contains('is-open')) setMenu(false);
|
||
});
|
||
|
||
window.addEventListener('resize', () => {
|
||
if (window.innerWidth > 720) setMenu(false);
|
||
else menu.setAttribute('aria-hidden', String(!nav.classList.contains('is-open')));
|
||
});
|
||
|
||
menu.setAttribute('aria-hidden', String(window.innerWidth <= 720));
|
||
})();
|
||
|
||
(() => {
|
||
const modal = document.querySelector('[data-console-modal]');
|
||
const panel = modal?.querySelector('.control-console-panel');
|
||
const openButtons = document.querySelectorAll('[data-console-open]');
|
||
const closeButtons = modal?.querySelectorAll('[data-console-close]') || [];
|
||
if (!modal || !panel || openButtons.length === 0) return;
|
||
|
||
function setConsole(open) {
|
||
modal.classList.toggle('is-open', open);
|
||
modal.setAttribute('aria-hidden', String(!open));
|
||
document.body.classList.toggle('has-control-console', open);
|
||
if (open) panel.focus();
|
||
}
|
||
|
||
openButtons.forEach((button) => {
|
||
button.addEventListener('click', () => setConsole(true));
|
||
});
|
||
|
||
closeButtons.forEach((button) => {
|
||
button.addEventListener('click', () => setConsole(false));
|
||
});
|
||
|
||
document.addEventListener('keydown', (event) => {
|
||
if (event.key === 'Escape' && modal.classList.contains('is-open')) setConsole(false);
|
||
});
|
||
})();
|
||
|
||
(() => {
|
||
const storageKey = 'site-theme';
|
||
const root = document.documentElement;
|
||
const button = document.querySelector('[data-theme-toggle]');
|
||
const media = window.matchMedia('(prefers-color-scheme: dark)');
|
||
const getSystemTheme = () => (media.matches ? 'dark' : 'light');
|
||
const getSavedTheme = () => {
|
||
try {
|
||
const theme = localStorage.getItem(storageKey);
|
||
return theme === 'dark' || theme === 'light' ? theme : null;
|
||
} catch {
|
||
return null;
|
||
}
|
||
};
|
||
const setSavedTheme = (theme) => {
|
||
try {
|
||
if (theme === getSystemTheme()) localStorage.removeItem(storageKey);
|
||
else localStorage.setItem(storageKey, theme);
|
||
} catch {
|
||
// localStorage can be unavailable in strict privacy modes.
|
||
}
|
||
};
|
||
const getTheme = () => getSavedTheme() || getSystemTheme();
|
||
|
||
function applyTheme(theme) {
|
||
root.dataset.theme = theme;
|
||
root.dataset.themeMode = getSavedTheme() ? 'manual' : 'auto';
|
||
button?.setAttribute('aria-label', theme === 'dark' ? '切换到日间模式' : '切换到夜间模式');
|
||
button?.setAttribute(
|
||
'title',
|
||
getSavedTheme()
|
||
? (theme === 'dark' ? '切换到日间模式' : '切换到夜间模式')
|
||
: `跟随系统：${theme === 'dark' ? '夜间' : '日间'}模式`
|
||
);
|
||
}
|
||
|
||
applyTheme(getTheme());
|
||
|
||
button?.addEventListener('click', () => {
|
||
const nextTheme = root.dataset.theme === 'dark' ? 'light' : 'dark';
|
||
setSavedTheme(nextTheme);
|
||
applyTheme(nextTheme);
|
||
});
|
||
|
||
const handleSystemThemeChange = () => {
|
||
if (!getSavedTheme()) applyTheme(getSystemTheme());
|
||
};
|
||
|
||
if (typeof media.addEventListener === 'function') {
|
||
media.addEventListener('change', handleSystemThemeChange);
|
||
} else if (typeof media.addListener === 'function') {
|
||
media.addListener(handleSystemThemeChange);
|
||
}
|
||
})();
|
||
|
||
(() => {
|
||
const runtime = document.getElementById('site-runtime');
|
||
if (!runtime) return;
|
||
|
||
const start = new Date(window.SITE_CONFIG.site.startTime).getTime();
|
||
const pad = (value) => String(value).padStart(2, '0');
|
||
|
||
function updateRuntime() {
|
||
const diff = Math.max(0, Date.now() - start);
|
||
const days = Math.floor(diff / 86400000);
|
||
const hours = Math.floor((diff / 3600000) % 24);
|
||
const minutes = Math.floor((diff / 60000) % 60);
|
||
const seconds = Math.floor((diff / 1000) % 60);
|
||
runtime.textContent = `${days} 天 ${pad(hours)} 时 ${pad(minutes)} 分 ${pad(seconds)} 秒`;
|
||
}
|
||
|
||
updateRuntime();
|
||
window.setInterval(updateRuntime, 1000);
|
||
})();
|
||
</script> </body> </html> <script>(function(){const postSlug = "7207243b";
|
||
|
||
(() => {
|
||
const article = document.querySelector('.post-article');
|
||
const progressBar = document.querySelector('[data-reading-progress]');
|
||
const resume = document.querySelector('[data-reading-resume]');
|
||
const resumePercent = document.querySelector('[data-reading-resume-percent]');
|
||
const continueButton = document.querySelector('[data-reading-continue]');
|
||
const restartButton = document.querySelector('[data-reading-restart]');
|
||
if (!article || !progressBar || !resume) return;
|
||
|
||
const storageKey = `reading-progress:${postSlug}`;
|
||
let saved = null;
|
||
let canPersist = true;
|
||
let frame = 0;
|
||
|
||
try {
|
||
saved = JSON.parse(localStorage.getItem(storageKey) || 'null');
|
||
} catch {
|
||
saved = null;
|
||
}
|
||
|
||
const articleMetrics = () => {
|
||
const top = article.getBoundingClientRect().top + window.scrollY;
|
||
const distance = Math.max(1, article.offsetHeight - window.innerHeight);
|
||
return { top, distance };
|
||
};
|
||
|
||
const getProgress = () => {
|
||
const { top, distance } = articleMetrics();
|
||
return Math.min(1, Math.max(0, (window.scrollY - top) / distance));
|
||
};
|
||
|
||
const persist = (progress) => {
|
||
if (!canPersist) return;
|
||
try {
|
||
if (progress >= 0.97) localStorage.removeItem(storageKey);
|
||
else if (progress >= 0.02) localStorage.setItem(storageKey, JSON.stringify({ progress, savedAt: Date.now() }));
|
||
} catch {
|
||
// Storage can be unavailable in strict privacy modes.
|
||
}
|
||
};
|
||
|
||
const update = () => {
|
||
frame = 0;
|
||
const progress = getProgress();
|
||
progressBar.style.transform = `scaleX(${progress})`;
|
||
persist(progress);
|
||
};
|
||
|
||
const requestUpdate = () => {
|
||
if (!frame) frame = window.requestAnimationFrame(update);
|
||
};
|
||
|
||
const closeResume = () => {
|
||
resume.hidden = true;
|
||
canPersist = true;
|
||
};
|
||
|
||
const savedIsFresh = saved && Date.now() - Number(saved.savedAt) < 90 * 86400000;
|
||
const savedProgress = Number(saved?.progress);
|
||
if (savedIsFresh && savedProgress >= 0.05 && savedProgress < 0.97) {
|
||
canPersist = false;
|
||
if (resumePercent) resumePercent.textContent = `${Math.round(savedProgress * 100)}%`;
|
||
window.setTimeout(() => { resume.hidden = false; }, 350);
|
||
}
|
||
|
||
continueButton?.addEventListener('click', () => {
|
||
closeResume();
|
||
const { top, distance } = articleMetrics();
|
||
window.scrollTo({ top: top + distance * savedProgress, behavior: 'smooth' });
|
||
});
|
||
|
||
restartButton?.addEventListener('click', () => {
|
||
try { localStorage.removeItem(storageKey); } catch {}
|
||
closeResume();
|
||
window.scrollTo({ top: Math.max(0, articleMetrics().top - 90), behavior: 'smooth' });
|
||
});
|
||
|
||
update();
|
||
window.addEventListener('scroll', requestUpdate, { passive: true });
|
||
window.addEventListener('resize', requestUpdate);
|
||
window.addEventListener('pagehide', () => persist(getProgress()));
|
||
})();
|
||
})();</script> <script>
|
||
(() => {
|
||
document.querySelectorAll('.article-outdated-note[data-freshness-at]').forEach((note) => {
|
||
const freshnessAt = new Date(note.dataset.freshnessAt || '');
|
||
const hasNewVersion = note.dataset.hasNewVersion === 'true';
|
||
if (Number.isNaN(freshnessAt.valueOf())) return;
|
||
|
||
const threshold = new Date();
|
||
threshold.setFullYear(threshold.getFullYear() - 1);
|
||
note.hidden = !hasNewVersion && freshnessAt >= threshold;
|
||
});
|
||
})();
|
||
</script> <script>
|
||
(() => {
|
||
const prose = document.querySelector('.prose[data-toc]');
|
||
if (!prose) return;
|
||
|
||
let toc = [];
|
||
try {
|
||
toc = JSON.parse(prose.dataset.toc || '[]');
|
||
} catch {
|
||
toc = [];
|
||
}
|
||
|
||
prose.querySelectorAll('h1, h2, h3').forEach((heading, index) => {
|
||
const item = toc[index];
|
||
if (item?.id && !heading.id) heading.id = item.id;
|
||
});
|
||
|
||
const tocTree = document.querySelector('[data-toc-tree]');
|
||
if (tocTree && toc.length > 0) {
|
||
const links = [...tocTree.querySelectorAll('[data-toc-link]')];
|
||
const groups = [...tocTree.querySelectorAll('[data-toc-group]')];
|
||
const headings = toc
|
||
.map((item) => document.getElementById(item.id))
|
||
.filter(Boolean);
|
||
|
||
const setActiveToc = (id) => {
|
||
links.forEach((link) => {
|
||
const isActive = link.getAttribute('href') === `#${id}`;
|
||
link.classList.toggle('is-active', isActive);
|
||
if (isActive) link.setAttribute('aria-current', 'location');
|
||
else link.removeAttribute('aria-current');
|
||
});
|
||
|
||
const activeLink = tocTree.querySelector(`[data-toc-link][href="#${CSS.escape(id)}"]`);
|
||
const activeGroup = activeLink?.closest('[data-toc-group]');
|
||
groups.forEach((group) => group.classList.toggle('is-open', group === activeGroup));
|
||
};
|
||
|
||
const updateToc = () => {
|
||
const anchorOffset = 132;
|
||
let activeId = headings[0]?.id;
|
||
|
||
headings.forEach((heading) => {
|
||
if (heading.getBoundingClientRect().top <= anchorOffset) activeId = heading.id;
|
||
});
|
||
|
||
if (activeId) setActiveToc(activeId);
|
||
};
|
||
|
||
tocTree.addEventListener('click', (event) => {
|
||
const link = event.target.closest?.('[data-toc-link]');
|
||
if (!link) return;
|
||
|
||
const group = link.closest('[data-toc-group]');
|
||
groups.forEach((item) => item.classList.toggle('is-open', item === group));
|
||
});
|
||
|
||
updateToc();
|
||
window.addEventListener('scroll', updateToc, { passive: true });
|
||
window.addEventListener('resize', updateToc);
|
||
}
|
||
|
||
})();
|
||
</script> <script>
|
||
(() => {
|
||
const prose = document.querySelector('.post-article .prose');
|
||
if (!prose) return;
|
||
|
||
const collapsedLineCount = 8;
|
||
const copyLabels = {
|
||
idle: '复制',
|
||
done: '已复制',
|
||
error: '复制失败',
|
||
};
|
||
|
||
const languageNames = {
|
||
bash: 'Bash',
|
||
c: 'C',
|
||
cpp: 'C++',
|
||
cs: 'C#',
|
||
css: 'CSS',
|
||
html: 'HTML',
|
||
js: 'JavaScript',
|
||
json: 'JSON',
|
||
jsx: 'JSX',
|
||
md: 'Markdown',
|
||
powershell: 'PowerShell',
|
||
py: 'Python',
|
||
shell: 'Shell',
|
||
sh: 'Shell',
|
||
ts: 'TypeScript',
|
||
tsx: 'TSX',
|
||
txt: 'Text',
|
||
yaml: 'YAML',
|
||
yml: 'YAML',
|
||
};
|
||
|
||
function getCodeLanguage(pre, code) {
|
||
const rawLanguage =
|
||
pre.dataset.language ||
|
||
code.dataset.language ||
|
||
[...pre.classList, ...code.classList]
|
||
.find((className) => className.startsWith('language-'))
|
||
?.replace(/^language-/, '') ||
|
||
'';
|
||
|
||
const normalized = rawLanguage.trim().toLowerCase();
|
||
return languageNames[normalized] || normalized.toUpperCase() || 'Code';
|
||
}
|
||
|
||
function getCodeText(code, lines) {
|
||
if (lines.length > 0) return lines.map((line) => line.textContent || '').join('\n');
|
||
return code.textContent || '';
|
||
}
|
||
|
||
prose.querySelectorAll('img').forEach((image) => {
|
||
if (image.closest('a, .article-header-cover')) return;
|
||
|
||
const src = image.currentSrc || image.getAttribute('src');
|
||
if (!src) return;
|
||
|
||
const link = document.createElement('a');
|
||
link.className = 'fancybox';
|
||
link.href = src;
|
||
link.dataset.fancybox = 'article-gallery';
|
||
link.dataset.caption = image.getAttribute('alt') || '';
|
||
image.parentNode?.insertBefore(link, image);
|
||
link.appendChild(image);
|
||
});
|
||
|
||
prose.querySelectorAll('table').forEach((table) => {
|
||
if (table.closest('.table-scroll')) return;
|
||
|
||
const wrapper = document.createElement('div');
|
||
wrapper.className = 'table-scroll';
|
||
table.parentNode?.insertBefore(wrapper, table);
|
||
wrapper.appendChild(table);
|
||
});
|
||
|
||
prose.querySelectorAll('.butterfly-hide-inline').forEach((item) => {
|
||
const button = item.querySelector('.butterfly-hide-toggle');
|
||
if (!button) return;
|
||
|
||
button.addEventListener('click', () => {
|
||
item.classList.toggle('is-open');
|
||
});
|
||
});
|
||
|
||
if (window.Fancybox) {
|
||
window.Fancybox.bind('[data-fancybox="article-gallery"]', {
|
||
compact: false,
|
||
dragToClose: true,
|
||
Images: {
|
||
zoom: true,
|
||
},
|
||
});
|
||
}
|
||
|
||
async function copyText(text) {
|
||
if (navigator.clipboard?.writeText) {
|
||
await navigator.clipboard.writeText(text);
|
||
return;
|
||
}
|
||
|
||
const textarea = document.createElement('textarea');
|
||
textarea.value = text;
|
||
textarea.setAttribute('readonly', '');
|
||
textarea.style.position = 'fixed';
|
||
textarea.style.top = '-9999px';
|
||
textarea.style.opacity = '0';
|
||
document.body.appendChild(textarea);
|
||
textarea.select();
|
||
document.execCommand('copy');
|
||
textarea.remove();
|
||
}
|
||
|
||
prose.querySelectorAll('pre').forEach((pre) => {
|
||
if (pre.closest('.code-block')) return;
|
||
|
||
const code = pre.querySelector('code');
|
||
if (!code) return;
|
||
|
||
const wrapper = document.createElement('div');
|
||
wrapper.className = 'code-block';
|
||
pre.parentNode?.insertBefore(wrapper, pre);
|
||
wrapper.appendChild(pre);
|
||
|
||
const header = document.createElement('div');
|
||
header.className = 'code-block-header';
|
||
|
||
const language = document.createElement('span');
|
||
language.className = 'code-block-language';
|
||
language.textContent = getCodeLanguage(pre, code);
|
||
|
||
const toolbar = document.createElement('div');
|
||
toolbar.className = 'code-block-toolbar';
|
||
|
||
const copyButton = document.createElement('button');
|
||
copyButton.className = 'code-block-copy';
|
||
copyButton.type = 'button';
|
||
copyButton.textContent = copyLabels.idle;
|
||
copyButton.setAttribute('aria-label', '复制代码');
|
||
|
||
const toggleButton = document.createElement('button');
|
||
toggleButton.className = 'code-block-toggle';
|
||
toggleButton.type = 'button';
|
||
toggleButton.textContent = '展开';
|
||
toggleButton.setAttribute('aria-expanded', 'false');
|
||
|
||
const expandCue = document.createElement('button');
|
||
expandCue.className = 'code-block-expand-cue';
|
||
expandCue.type = 'button';
|
||
expandCue.setAttribute('aria-label', 'Expand code block');
|
||
expandCue.setAttribute('aria-expanded', 'false');
|
||
expandCue.innerHTML = '<span aria-hidden="true"></span>';
|
||
|
||
toolbar.append(copyButton, toggleButton);
|
||
header.append(language, toolbar);
|
||
wrapper.appendChild(header);
|
||
wrapper.appendChild(expandCue);
|
||
|
||
const codeLines = [...code.querySelectorAll(':scope > .line')];
|
||
if (codeLines.length > 0) {
|
||
wrapper.classList.add('has-line-numbers');
|
||
wrapper.style.setProperty('--line-number-width', `${String(codeLines.length).length + 1}ch`);
|
||
codeLines.forEach((line, index) => {
|
||
line.dataset.line = String(index + 1);
|
||
});
|
||
}
|
||
|
||
const lineCount = codeLines.length || (code.textContent || '').split(/\r?\n/).length;
|
||
if (lineCount > collapsedLineCount) wrapper.classList.add('is-collapsible');
|
||
|
||
copyButton.addEventListener('click', async () => {
|
||
try {
|
||
await copyText(getCodeText(code, codeLines));
|
||
copyButton.textContent = copyLabels.done;
|
||
window.setTimeout(() => {
|
||
copyButton.textContent = copyLabels.idle;
|
||
}, 1400);
|
||
} catch {
|
||
copyButton.textContent = copyLabels.error;
|
||
window.setTimeout(() => {
|
||
copyButton.textContent = copyLabels.idle;
|
||
}, 1600);
|
||
}
|
||
});
|
||
|
||
function toggleExpanded() {
|
||
const isExpanded = wrapper.classList.toggle('is-expanded');
|
||
toggleButton.textContent = isExpanded ? '收起' : '展开';
|
||
toggleButton.setAttribute('aria-expanded', String(isExpanded));
|
||
expandCue.setAttribute('aria-expanded', String(isExpanded));
|
||
}
|
||
|
||
toggleButton.addEventListener('click', toggleExpanded);
|
||
expandCue.addEventListener('click', toggleExpanded);
|
||
});
|
||
})();
|
||
</script> |