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README.html
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<!DOCTYPE html><html><head>
<title>README</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="stylesheet" href="file:///c:\Users\woodt\.vscode\extensions\shd101wyy.markdown-preview-enhanced-0.5.0\node_modules\@shd101wyy\mume\dependencies\katex\katex.min.css">
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<body for="html-export">
<div class="mume markdown-preview ">
<h1 class="mume-header" id="ray-tracing-in-one-weekend">Ray Tracing in one Weekend</h1>
<h2 class="mume-header" id="%E5%85%B3%E4%BA%8E%E6%9C%AC%E6%95%99%E7%A8%8B">关于本教程</h2>
<p>此教程的原版是 Peter Shirley 的 <a href="http://www.realtimerendering.com/raytracing/Ray%20Tracing%20in%20a%20Weekend.pdf">"Ray Tracing in One Weekend"</a> . 他把Ray tracing讲的深入浅出，水平很高，用来入门是很好的教程。但是原文是英文的，书中的例子是用C++写的，所以需要一定的C++和英文基础。本教程是用Unity把原教程的例子用C#重新写了一遍，所以特别适合Unity 3D的程序员。其实也可不用Unity，主要是这样运行例子比较简单，省的安装一些其他类库。<br>
本教程是用markdown写的。因为嵌入了tex，可能不同的浏览器看到的不大一样。如果看到数学公式解析有问题，就看README.pdf，在readme.md的同级目录里面。</p>
<h2 class="mume-header" id="%E5%85%B3%E4%BA%8E%E4%BB%A3%E7%A0%81">关于代码</h2>
<p>运行：项目代码包含在Unity的工程里面，可以用Unity打开在Raytracing菜单,点击相应的章节就可以运行代码。<br>
代码冗余的说明：不同章节的代码里有些冗余的类，是因为在不同的章节中可能会少有改动。就让不同的章节都包含了一份完整的代码。<br>
代码运行时间：后面几章的代码运行时间会比较久，尤其是最后一张可能会有几个小时。如果想快速看到结果，可以修改生成图片的大小和采样数。<br>
Unity版本：对Unity版本没有特殊要求，在Unity2018.4 和Unity2018.3上测试过，其他版本应该问题不大</p>
<h2 class="mume-header" id="%E5%85%B3%E4%BA%8Eray-tracing">关于Ray tracing</h2>
<p>Ray traing是一种计算机渲染图像的算法，特点是计算量大但是效果好，之前主要用于电影或者动画的渲染。Nvidia的显卡支持了实时的Ray tracing，现在也可以在实时渲染的地方用了。之前实时的图形学使用的是光栅化算法。关于光栅化的算法介绍，推荐一个<a href="http://www.scratchapixel.com/lessons/3d-basic-rendering/rasterization-practical-implementation">网站</a>上面讲的很详细。</p>
<h2 class="mume-header" id="%E7%AC%AC%E4%B8%80%E7%AB%A0-%E8%BE%93%E5%87%BA%E4%B8%80%E5%BC%A0%E5%9B%BE%E7%89%87">第一章 输出一张图片</h2>
<p>这章讲一下，计算机中图片，图片是由像素构成的，每个像素，可以被编码成一个[R,G,B]的向量。关于RGB和颜色空间，也推荐一个<a href="http://www.scratchapixel.com/lessons/digital-imaging/colors">网页</a>上面有比较详细的介绍。看下面这段代码生成了一张图片</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Chapter1</span>
<span class="token punctuation">{</span>
<span class="token punctuation">[</span><span class="token class-name">MenuItem</span><span class="token punctuation">(</span><span class="token string">"Raytracing/Chapter1"</span><span class="token punctuation">)</span><span class="token punctuation">]</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">Main</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> nx <span class="token operator">=</span> <span class="token number">1280</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> ny <span class="token operator">=</span> <span class="token number">720</span><span class="token punctuation">;</span>
<span class="token class-name">Texture2D</span> tex <span class="token operator">=</span> ImageHelper<span class="token punctuation">.</span><span class="token function">CreateImg</span><span class="token punctuation">(</span>nx<span class="token punctuation">,</span> ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> ny <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">>=</span> <span class="token number">0</span><span class="token punctuation">;</span> <span class="token operator">--</span>j<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> nx<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">float</span> r <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>i<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>nx<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> g <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>j<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> b <span class="token operator">=</span> <span class="token number">0.2f</span><span class="token punctuation">;</span>
ImageHelper<span class="token punctuation">.</span><span class="token function">SetPixel</span><span class="token punctuation">(</span>tex<span class="token punctuation">,</span> i<span class="token punctuation">,</span> j<span class="token punctuation">,</span> r<span class="token punctuation">,</span> g<span class="token punctuation">,</span> b<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
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ImageHelper<span class="token punctuation">.</span><span class="token function">SaveImg</span><span class="token punctuation">(</span>tex<span class="token punctuation">,</span> <span class="token string">"img\chapter1.png"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
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<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">class</span> <span class="token class-name">ImageHelper</span>
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Texture2D</span> <span class="token function">CreateImg</span><span class="token punctuation">(</span><span class="token keyword">int</span> width <span class="token punctuation">,</span><span class="token keyword">int</span> height<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Texture2D</span> tex <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Texture2D</span><span class="token punctuation">(</span>width<span class="token punctuation">,</span> height<span class="token punctuation">,</span> TextureFormat<span class="token punctuation">.</span>RGB24<span class="token punctuation">,</span> <span class="token keyword">false</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> tex<span class="token punctuation">;</span>
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<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">SetPixel</span><span class="token punctuation">(</span><span class="token class-name">Texture2D</span> tex<span class="token punctuation">,</span><span class="token keyword">int</span> x<span class="token punctuation">,</span><span class="token keyword">int</span> y<span class="token punctuation">,</span><span class="token keyword">float</span> r<span class="token punctuation">,</span><span class="token keyword">float</span> g<span class="token punctuation">,</span><span class="token keyword">float</span> b<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
tex<span class="token punctuation">.</span><span class="token function">SetPixel</span><span class="token punctuation">(</span>x<span class="token punctuation">,</span> y<span class="token punctuation">,</span> <span class="token keyword">new</span> <span class="token class-name">Color</span><span class="token punctuation">(</span>r<span class="token punctuation">,</span> g<span class="token punctuation">,</span> b<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">SetPixel</span><span class="token punctuation">(</span><span class="token class-name">Texture2D</span> tex<span class="token punctuation">,</span><span class="token keyword">int</span> x<span class="token punctuation">,</span><span class="token keyword">int</span> y<span class="token punctuation">,</span><span class="token class-name">Vector3</span> color<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
tex<span class="token punctuation">.</span><span class="token function">SetPixel</span><span class="token punctuation">(</span>x<span class="token punctuation">,</span> y<span class="token punctuation">,</span> <span class="token keyword">new</span> <span class="token class-name">Color</span><span class="token punctuation">(</span>color<span class="token punctuation">.</span>x<span class="token punctuation">,</span> color<span class="token punctuation">.</span>y<span class="token punctuation">,</span> color<span class="token punctuation">.</span>z<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">SaveImg</span><span class="token punctuation">(</span><span class="token class-name">Texture2D</span> tex<span class="token punctuation">,</span> <span class="token keyword">string</span> path<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">var</span> bytes <span class="token operator">=</span> tex<span class="token punctuation">.</span><span class="token function">EncodeToPNG</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
File<span class="token punctuation">.</span><span class="token function">WriteAllBytes</span><span class="token punctuation">(</span>Path<span class="token punctuation">.</span><span class="token function">Combine</span><span class="token punctuation">(</span>Application<span class="token punctuation">.</span>dataPath<span class="token punctuation">,</span> path<span class="token punctuation">)</span><span class="token punctuation">,</span> bytes<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>在Unity的菜单中运行后，得到如下结果，保存在Img文件夹chapter1.png。<img src="Assets/Img/chapter1.png" alt="alt text" title="chapter1">。ImageHelper直接使用了Unity中保存图片的接口。Main函数中就是对一张1280 X 720的每个像素点赋值。这样我们就得到了一张用算法生成的图片。</p>
<h2 class="mume-header" id="%E7%AC%AC%E4%BA%8C%E7%AB%A0-%E5%90%91%E9%87%8F%E7%B1%BB">第二章 向量类</h2>
<p>这章主要介绍向量的基本知识。如果有不熟悉的，请在网上查一下。这里就不细讲了。因为使用了Unity，里面有自带的向量类Vector3 ，所以在此教程中就不自己实现了。</p>
<h2 class="mume-header" id="%E7%AC%AC%E4%B8%89%E7%AB%A0-%E5%B0%84%E7%BA%BF%E7%AE%80%E5%8D%95%E7%9A%84%E6%91%84%E5%83%8F%E6%9C%BA%E5%92%8C%E8%83%8C%E6%99%AF">第三章 射线，简单的摄像机和背景</h2>
<p>ray tracing算法，有翻译为光线追踪的，也有翻译为射线追踪,所以肯定要用到射线。本书里面的射线是由参数方程表示的，射线是由一个端点和一个方向向量组成的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>+</mo><mi>t</mi><mo>∗</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">p(t) = A + t * \vec B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span> 。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span></span></span></span>就是三维空间中的一个点，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>B</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span>是射线的方向是一个向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>是一个实数。当<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>从0变化到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">+\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span>时，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>取遍射线上所有的点。如果<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span> 取负数，那么<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>可以理解为跟原来射线同原点但是方向相反的射线。上一张原教程中的图：<img src="Assets/figure/chapter3-1.png" alt="alt text"><br>
要实现一个Ray的类很简单，而且Unity中实现了一个Ray的类，就不自己实现了。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">namespace</span> UnityEngine
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token keyword">struct</span> Ray
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token function">Ray</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> origin<span class="token punctuation">,</span> <span class="token class-name">Vector3</span> direction<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> origin <span class="token punctuation">{</span> <span class="token keyword">get</span><span class="token punctuation">;</span> <span class="token keyword">set</span><span class="token punctuation">;</span> <span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> direction <span class="token punctuation">{</span> <span class="token keyword">get</span><span class="token punctuation">;</span> <span class="token keyword">set</span><span class="token punctuation">;</span> <span class="token punctuation">}</span>
<span class="token comment">//</span>
<span class="token comment">// 摘要:</span>
<span class="token comment">// Returns a point at distance units along the ray.</span>
<span class="token comment">//</span>
<span class="token comment">// 参数:</span>
<span class="token comment">// distance:</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> <span class="token function">GetPoint</span><span class="token punctuation">(</span><span class="token keyword">float</span> distance<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>要得到射线指向的点，即射线的终点，我们需要计算<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span> 就是使用GetPoint方法，t可以理解为从与原点发射出去的距离distance。有兴趣的可以自己写一个Ray的类实现一下，看看跟Unity的计算结果是否相同。</p>
<p>有了这些基础知识，我们现在就可以写一个简单的Ray tracer了。先让我们来想一下，相机是如何拍照的的，假设镜头是一个小孔，无数的光线从这里穿过，最终在底片上留下的痕迹，形成照片。因为光线是可逆的，我们可以假想，光线是从最后的照片的某个像素出发，然后穿过镜头，与真实世界里面的物体发生作用，通过遇到的物体来决定改像素的颜色，这个就是Ray tracer的算法的核心思想。由于相机的参数一旦确定，如下图：<br>
<img src="Assets/figure/chapter3-2.png" alt="alt text"><br>
我们就可以忽略掉底片，假想光线是从相机直接投在前面的幕布上的。然后我们来写第一个光线追踪算法：<br>
<img src="Assets/figure/chapter3-3.png" alt="alt text"><br>
假如我们的以摄像机所在的位置的作为原点，用右手坐标系，幕布就在Z轴的负方向上，且与XY屏幕平行。幕布的左上角在空间的（-2，1，-1），右下角在（2，-1，-1）（这里我们假设，幕布的分辨率是1280 X 720的）现在我们从相机发射一道射线，指向幕布上的一个像素点，这样就得到了条射线，然后我们根据射线的方向向量，来插值两个颜色，最终得到像素的颜色。等射线跑遍了整个幕布，这是我们就得到了一张图。这里用射线的Y方向做差值，图像就是从上到下渐变的，没有什么道理只是单纯的图片不那么单调。代码如下：</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Chapter3</span>
<span class="token punctuation">{</span>
<span class="token keyword">private</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> topColor <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>one<span class="token punctuation">;</span>
<span class="token keyword">private</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> bottomColor <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0.5f</span><span class="token punctuation">,</span> <span class="token number">0.7f</span><span class="token punctuation">,</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> unit_direction <span class="token operator">=</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">float</span> t <span class="token operator">=</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>unit_direction<span class="token punctuation">.</span>y <span class="token operator">+</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span><span class="token function">Lerp</span><span class="token punctuation">(</span>topColor<span class="token punctuation">,</span> bottomColor<span class="token punctuation">,</span> t<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">[</span><span class="token class-name">MenuItem</span><span class="token punctuation">(</span><span class="token string">"Raytracing/Chapter3"</span><span class="token punctuation">)</span><span class="token punctuation">]</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">Main</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> nx <span class="token operator">=</span> <span class="token number">1280</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> ny <span class="token operator">=</span> <span class="token number">720</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> lower_left_corner <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">2.0f</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1.0f</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> horizontal <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">4.0f</span><span class="token punctuation">,</span> <span class="token number">0.0f</span><span class="token punctuation">,</span> <span class="token number">0.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> vertical <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0.0f</span><span class="token punctuation">,</span> <span class="token number">2.0f</span><span class="token punctuation">,</span> <span class="token number">0.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> origin <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>zero<span class="token punctuation">;</span>
<span class="token class-name">Texture2D</span> tex <span class="token operator">=</span> ImageHelper<span class="token punctuation">.</span><span class="token function">CreateImg</span><span class="token punctuation">(</span>nx<span class="token punctuation">,</span> ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> ny <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">>=</span> <span class="token number">0</span><span class="token punctuation">;</span> <span class="token operator">--</span>j<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> nx<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">float</span> u <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>i<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>nx<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> v <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>j<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Ray</span> r <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Ray</span><span class="token punctuation">(</span>origin<span class="token punctuation">,</span> lower_left_corner <span class="token operator">+</span> u <span class="token operator">*</span> horizontal <span class="token operator">+</span> v <span class="token operator">*</span> vertical<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> color <span class="token operator">=</span> <span class="token function">RayCast</span><span class="token punctuation">(</span>r<span class="token punctuation">)</span><span class="token punctuation">;</span>
ImageHelper<span class="token punctuation">.</span><span class="token function">SetPixel</span><span class="token punctuation">(</span>tex<span class="token punctuation">,</span> i<span class="token punctuation">,</span> j<span class="token punctuation">,</span> color<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
ImageHelper<span class="token punctuation">.</span><span class="token function">SaveImg</span><span class="token punctuation">(</span>tex<span class="token punctuation">,</span> <span class="token string">"Assets/Img/chapter3.png"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>最终渲染的图片如下：<img src="Assets/Img/chapter3.png" alt="alt text" title="chapter3"></p>
<h2 class="mume-header" id="%E7%AC%AC%E5%9B%9B%E7%AB%A0-%E7%94%BB%E4%B8%AA%E5%9C%86">第四章 画个圆</h2>
<p>这次我们在场景里添加个圆，为啥是个圆呢，因为判断射线跟圆是否相交最简单。圆的方程： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>c</mi><mi>y</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>c</mi><mi>z</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(x-cx)^2 + (y-cy)^2 + (z-cz)^2 = R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 这里 (cx,cy,cz) 是圆的中心点坐标，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span> 是半径。其实这个方程就是描述了圆的几何性质：圆上任意一点到圆心的距离是相等的。如果写成向量的形式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>c</mi><mi>x</mi><mo separator="true">,</mo><mi>c</mi><mi>y</mi><mo separator="true">,</mo><mi>c</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vec C = (cx,cy,cz)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> 是圆心点, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>P</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">\vec P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span>是 圆上任意一点,<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span>是半径，那么圆的方程可以写成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∥</mi><mover accent="true"><mi>P</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mi mathvariant="normal">∥</mi><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Vert \vec P - \vec C \Vert = R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord">∥</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mord">∥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> ，也就是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>P</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>P</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">dot(\vec P - \vec C,\vec P - \vec C) = R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1607699999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 这里<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>o</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">dot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span></span></span></span>代表向量的点乘。还记得射线的向量形式么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>+</mo><mi>t</mi><mo>∗</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">p(t) = A + t * \vec B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span>。如果射线与圆有交点的话，也就是射线上存在一个点，可以满足圆的方程式，因为那个点也在圆上。所以这里我们把射线的方程带入到圆面方程里面，得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">dot(p(t) - \vec C,p(t) - \vec C) = R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 也就是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>+</mo><mi>t</mi><mo>∗</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>+</mo><mi>t</mi><mo>∗</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msup><mi>R</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">dot (\vec A + t * \vec B - \vec C,\vec A + t * \vec B - \vec C) = R^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0496599999999998em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1607699999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0496599999999998em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 这里把点乘展开，并整理可得<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>∗</mo><mi>t</mi><mo>∗</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo>∗</mo><mi>t</mi><mo>∗</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>R</mi><mo>∗</mo><mi>R</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t*t*dot(\vec B,\vec B) + 2*t*dot(\vec B,\vec A - \vec C) + dot(\vec A - \vec C,\vec A -\vec C) - R*R = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1607699999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> 这里看上去有点复杂，其实除了t剩下都是常量，假如让<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>b</mi><mo>=</mo><mn>2</mn><mo>∗</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>B</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>c</mi><mo>=</mo><mi>d</mi><mi>o</mi><mi>t</mi><mo stretchy="false">(</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>A</mi><mo>⃗</mo></mover><mo>−</mo><mover accent="true"><mi>C</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>R</mi><mo>∗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a = dot(\vec B,\vec B),b = 2 * dot(\vec B,\vec A - \vec C), c = dot(\vec A - \vec C,\vec A -\vec C) - R*R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.05017em;">B</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1607699999999999em;vertical-align:-0.19444em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault">A</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.09660999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.21633em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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如果方程有两个根那就是射线与圆相交，如果有一个根那就是射线与圆相切，如果一个根都没有就是不相交。还记得一元二次方程根的公式么，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mo>±</mo><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.384482em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.039482em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.3939999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathdefault mtight">b</span><span class="mbin mtight">±</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9221171428571429em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">4</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">c</span></span></span><span style="top:-2.882117142857143em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.11788285714285718em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，有没有根就看<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">b^2 - 4ac \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">4</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></p>
<p>这样我们在场景里面添加一个圆，位置为(0,0,-1) 半径是0.5的圆。做Raycast的时候，如果射线跟圆相交，直接返回红色，不相交还是跟上一章一样。下面就是与上一章不同的代码部分，完整的代码在chapter4.cs中。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"> <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">bool</span> <span class="token function">Hit_sphere</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> center<span class="token punctuation">,</span><span class="token keyword">float</span> radius<span class="token punctuation">,</span><span class="token class-name">Ray</span> ray<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> oc <span class="token operator">=</span> ray<span class="token punctuation">.</span>origin <span class="token operator">-</span> center<span class="token punctuation">;</span>
<span class="token keyword">float</span> a <span class="token operator">=</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>ray<span class="token punctuation">.</span>direction<span class="token punctuation">,</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> b <span class="token operator">=</span> <span class="token number">2.0f</span> <span class="token operator">*</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>oc<span class="token punctuation">,</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> c <span class="token operator">=</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>oc<span class="token punctuation">,</span> oc<span class="token punctuation">)</span> <span class="token operator">-</span> radius <span class="token operator">*</span> radius<span class="token punctuation">;</span>
<span class="token keyword">float</span> d <span class="token operator">=</span> b <span class="token operator">*</span> b <span class="token operator">-</span> <span class="token number">4</span> <span class="token operator">*</span> a <span class="token operator">*</span> c<span class="token punctuation">;</span>
<span class="token keyword">return</span> d <span class="token operator">></span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span><span class="token function">Hit_sphere</span><span class="token punctuation">(</span>center<span class="token punctuation">,</span> radius<span class="token punctuation">,</span> ray<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> ballColor<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token class-name">Vector3</span> unit_direction <span class="token operator">=</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">float</span> t <span class="token operator">=</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>unit_direction<span class="token punctuation">.</span>y <span class="token operator">+</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span><span class="token function">Lerp</span><span class="token punctuation">(</span>topColor<span class="token punctuation">,</span> bottomColor<span class="token punctuation">,</span> t<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</pre><p>渲染结果如下：<br>
<img src="Assets/Img/chapter4.png" alt="alt text" title="chapter4"><br>
虽然很丑，但是确实是渲染出来东西了。这里只计算了是否碰到了球，就统一给了个颜色，没有做着色，也没有计算光线遇到物体后的反射。后面让我们一点一点加上。</p>
<h2 class="mume-header" id="%E7%AC%AC%E4%BA%94%E7%AB%A0-%E5%9C%86%E7%9A%84%E8%A1%A8%E9%9D%A2%E6%B3%95%E7%BA%BF%E5%92%8C%E5%A4%9A%E4%B8%AA%E5%9C%86%E7%9A%84%E6%B8%B2%E6%9F%93">第五章 圆的表面法线和多个圆的渲染</h2>
<p>首先，我们来讲一下什么是物体的法线，法线就是一个垂直于物体表面的向量。圆的法线很容易计算，用圆上任意一点的坐标，减去圆心的坐标，得到的向量就是该点的法向量。在应用中我们会使用法向量的单位向量，（单位向量就是方向和法向量一样，但是长度是1的向量）这样可以避免计算中的很多数值问题。 如下图：<br>
<img src="Assets/figure/figure5-1.png" alt="alt text"><br>
有了法向量，我们把它用在上次渲染的地方，现在的问题就是如果把单位的法向量变成一个颜色值。单位法向量的三个分量取值范围都是[-1,1]之间，这样我们把三个分量分别加上1，然后再乘以0.5,这样的三个分量就落在了[0,1]的区间，又正好是三个分量，我们直接把变换后的x,y,z 三个分量对于 r,g,b的颜色分量，这样就把一个单位向量编码成了一个颜色值。看上去是不是有点随意，其实实时渲染中的Normal Map就是这样编码的。代码在chapter5_1.cs中，这里只贴跟上次不一样的地方。之前Hit_sphere函数只是返回圆是否和射线相交，这里会返回一个真正的根，这样就可以得到交点的具体坐标，从而计算出交点的法向量，然后用上面提到的方法，编码成一个颜色向量。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"> <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">float</span> <span class="token function">Hit_sphere</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> center<span class="token punctuation">,</span><span class="token keyword">float</span> radius<span class="token punctuation">,</span><span class="token class-name">Ray</span> ray<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> oc <span class="token operator">=</span> ray<span class="token punctuation">.</span>origin <span class="token operator">-</span> center<span class="token punctuation">;</span>
<span class="token keyword">float</span> a <span class="token operator">=</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>ray<span class="token punctuation">.</span>direction<span class="token punctuation">,</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> b <span class="token operator">=</span> <span class="token number">2.0f</span> <span class="token operator">*</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>oc<span class="token punctuation">,</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> c <span class="token operator">=</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>oc<span class="token punctuation">,</span> oc<span class="token punctuation">)</span> <span class="token operator">-</span> radius <span class="token operator">*</span> radius<span class="token punctuation">;</span>
<span class="token keyword">float</span> d <span class="token operator">=</span> b <span class="token operator">*</span> b <span class="token operator">-</span> <span class="token number">4</span> <span class="token operator">*</span> a <span class="token operator">*</span> c<span class="token punctuation">;</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>d <span class="token operator"><</span> <span class="token number">0</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">else</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> <span class="token punctuation">(</span><span class="token operator">-</span>b <span class="token operator">-</span> Mathf<span class="token punctuation">.</span><span class="token function">Sqrt</span><span class="token punctuation">(</span>d<span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> a<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">var</span> t <span class="token operator">=</span> <span class="token function">Hit_sphere</span><span class="token punctuation">(</span>center<span class="token punctuation">,</span> radius<span class="token punctuation">,</span> ray<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>t <span class="token operator">></span> <span class="token number">0</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> N <span class="token operator">=</span> <span class="token punctuation">(</span>ray<span class="token punctuation">.</span><span class="token function">GetPoint</span><span class="token punctuation">(</span>t<span class="token punctuation">)</span> <span class="token operator">-</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">return</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>N <span class="token operator">+</span> Vector3<span class="token punctuation">.</span>one<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token class-name">Vector3</span> unit_direction <span class="token operator">=</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
t <span class="token operator">=</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>unit_direction<span class="token punctuation">.</span>y <span class="token operator">+</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span><span class="token function">Lerp</span><span class="token punctuation">(</span>topColor<span class="token punctuation">,</span> bottomColor<span class="token punctuation">,</span> t<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</pre><p>渲染结果如下：<br>
<img src="Assets/Img/chapter5_1.png" alt="alt text" title="chapter5_1"></p>
<p>下面我们来解决如何渲染多个圆的问题,其实也很简单，做法比较简单粗暴，就是用射线跟每一个球做碰撞，找出那个交点最近就行了。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">struct</span> Hit_record
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token keyword">float</span> t<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> hitpoint<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> normal<span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">interface</span> <span class="token class-name">Hitable</span>
<span class="token punctuation">{</span>
<span class="token keyword">bool</span> <span class="token function">Hit</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> r<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_min<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_max<span class="token punctuation">,</span> <span class="token keyword">out</span> <span class="token class-name">Hit_record</span> record<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Sphere</span> <span class="token punctuation">:</span> <span class="token class-name">Hitable</span>
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> center<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">float</span> radius<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token function">Sphere</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
center <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>zero<span class="token punctuation">;</span>
radius <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token function">Sphere</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> c<span class="token punctuation">,</span> <span class="token keyword">float</span> r<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
center <span class="token operator">=</span> c<span class="token punctuation">;</span>
radius <span class="token operator">=</span> r<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">bool</span> <span class="token function">Hit</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_min<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_max<span class="token punctuation">,</span> <span class="token keyword">out</span> <span class="token class-name">Hit_record</span> record<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token comment">//具体内容见chapter5_2.cs</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>这里Hit_record 用来记录交点的信息。Hitable是个抽象类，为了将来场景里面能渲染别的东西。我们把圆的一下方法类封装了变成一个Sphere类，这里判断碰撞的时候，传入了参数的最大最小范围，如果不范围内，这样就当是没有交点那样处理</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">HitList</span> <span class="token punctuation">:</span> <span class="token class-name">Hitable</span>
<span class="token punctuation">{</span>
<span class="token keyword">private</span> List<span class="token operator"><</span>Hitable<span class="token operator">></span> list <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token generic-method"><span class="token function">List</span><span class="token punctuation"><</span><span class="token class-name">Hitable</span><span class="token punctuation">></span></span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token function">HitList</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">int</span> <span class="token function">GetCount</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> list<span class="token punctuation">.</span>Count<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">void</span> <span class="token function">Add</span><span class="token punctuation">(</span><span class="token class-name">Hitable</span> item<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
list<span class="token punctuation">.</span><span class="token function">Add</span><span class="token punctuation">(</span>item<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">bool</span> <span class="token function">Hit</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> r<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_min<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token keyword">float</span> t_max<span class="token punctuation">,</span> <span class="token keyword">out</span> <span class="token class-name">Hit_record</span> record<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Hit_record</span> temp_rec <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Hit_record</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
record <span class="token operator">=</span> temp_rec<span class="token punctuation">;</span>
<span class="token keyword">bool</span> hit_anything <span class="token operator">=</span> <span class="token keyword">false</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> closest_so_far <span class="token operator">=</span> t_max<span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> list<span class="token punctuation">.</span>Count<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>list<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">.</span><span class="token function">Hit</span><span class="token punctuation">(</span>r<span class="token punctuation">,</span> <span class="token keyword">ref</span> t_min<span class="token punctuation">,</span> <span class="token keyword">ref</span> closest_so_far<span class="token punctuation">,</span> <span class="token keyword">out</span> temp_rec<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
hit_anything <span class="token operator">=</span> <span class="token keyword">true</span><span class="token punctuation">;</span>
closest_so_far <span class="token operator">=</span> temp_rec<span class="token punctuation">.</span>t<span class="token punctuation">;</span>
record <span class="token operator">=</span> temp_rec<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">return</span> hit_anything<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>HitList 用来存放场景里面所有可以渲染的物体，每次做射线检测的时候，就遍历一遍场景里面的物体，找出最近的碰撞点。剩下的代码可以参加chapter5_2.cs<br>
最后渲染的结果为<img src="Assets/Img/chapter5_2.png" alt="alt text" title="chapter5_2"><br>
下面那个绿色的地面，其实是一个半径很大的圆。</p>
<h2 class="mume-header" id="%E7%AC%AC%E5%85%AD%E7%AB%A0-%E5%8F%8D%E8%B5%B0%E6%A0%B7">第六章 反走样</h2>
<p>将上一张最后的图片放大，可以看到球的边缘，有很多锯齿。这个是因为，我们是用一根光线碰撞的结果代表了整个像素的值，实际上一个像素上可能是无数光线共同作用的结果。所以这里我们模拟这个过程，用近似的方法出来每个像素。我们在该像素上，随机产生一点偏离，保证光线还在像素格子里，这个发射多根光线，最好对结果做一个平均值。具体的代码在chapter6.cs中</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">Main</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> nx <span class="token operator">=</span> <span class="token number">1280</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> ny <span class="token operator">=</span> <span class="token number">640</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> ns <span class="token operator">=</span> <span class="token number">64</span><span class="token punctuation">;</span>
<span class="token class-name">RayCamera</span> camera <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">RayCamera</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">HitList</span> list <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">HitList</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
list<span class="token punctuation">.</span><span class="token function">Add</span><span class="token punctuation">(</span><span class="token keyword">new</span> <span class="token class-name">Sphere</span><span class="token punctuation">(</span><span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token number">0.5f</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
list<span class="token punctuation">.</span><span class="token function">Add</span><span class="token punctuation">(</span><span class="token keyword">new</span> <span class="token class-name">Sphere</span><span class="token punctuation">(</span><span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">100.5f</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">,</span> <span class="token number">100</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Texture2D</span> tex <span class="token operator">=</span> ImageHelper<span class="token punctuation">.</span><span class="token function">CreateImg</span><span class="token punctuation">(</span>nx<span class="token punctuation">,</span> ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> ny <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">>=</span> <span class="token number">0</span><span class="token punctuation">;</span> <span class="token operator">--</span>j<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> nx<span class="token punctuation">;</span> <span class="token operator">++</span>i<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> color <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>zero<span class="token punctuation">;</span>
<span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> k <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> k <span class="token operator"><</span> ns<span class="token punctuation">;</span> <span class="token operator">++</span>k<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">float</span> u <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>i <span class="token operator">+</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>nx<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> v <span class="token operator">=</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>j <span class="token operator">+</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>ny<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Ray</span> r <span class="token operator">=</span> camera<span class="token punctuation">.</span><span class="token function">GetRay</span><span class="token punctuation">(</span>u<span class="token punctuation">,</span> v<span class="token punctuation">)</span><span class="token punctuation">;</span>
color <span class="token operator">+=</span> <span class="token function">RayCast</span><span class="token punctuation">(</span>r<span class="token punctuation">,</span> list<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
color <span class="token operator">=</span> color <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token punctuation">(</span>ns<span class="token punctuation">)</span><span class="token punctuation">;</span>
ImageHelper<span class="token punctuation">.</span><span class="token function">SetPixel</span><span class="token punctuation">(</span>tex<span class="token punctuation">,</span> i<span class="token punctuation">,</span> j<span class="token punctuation">,</span> color<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>这里ns的值就是一个像素采样的次数，当然采样的次数越多，效果越好，不过计算量也越大。运行这段代码的时候，需要等一会。可以算一下计算量有多大，这里一共有1280 X 640的像素，<br>
一个像素需要采样64次，一次采样需要解2个一元二次方程组。也就是说，一共需要解1280X640X64X2=104857600个一元二次方程组。如果不想等那么久可以简单把图片改小一点。这次我们用到了RayCamera这个类<br>
具体的实现在camera.cs中,也就是简单的封装了一下之前的方法。通过幕布的位置确定了摄像机，封装了GetRay方法，通过幕布的uv，来获得一根射线。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">RayCamera</span>
<span class="token punctuation">{</span>
<span class="token keyword">private</span> <span class="token class-name">Vector3</span> origin<span class="token punctuation">;</span>
<span class="token keyword">private</span> <span class="token class-name">Vector3</span> lower_left_corner<span class="token punctuation">;</span>
<span class="token keyword">private</span> <span class="token class-name">Vector3</span> horizontal<span class="token punctuation">;</span>
<span class="token keyword">private</span> <span class="token class-name">Vector3</span> vertical<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token function">RayCamera</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
origin <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>zero<span class="token punctuation">;</span>
lower_left_corner <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">2.0f</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1.0f</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
horizontal <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
vertical <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token function">RayCamera</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> ori<span class="token punctuation">,</span><span class="token class-name">Vector3</span> corner<span class="token punctuation">,</span><span class="token class-name">Vector3</span> h<span class="token punctuation">,</span><span class="token class-name">Vector3</span> v<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
origin <span class="token operator">=</span> ori<span class="token punctuation">;</span>
lower_left_corner <span class="token operator">=</span> corner<span class="token punctuation">;</span>
horizontal <span class="token operator">=</span> h<span class="token punctuation">;</span>
vertical <span class="token operator">=</span> v<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token class-name">Ray</span> <span class="token function">GetRay</span><span class="token punctuation">(</span><span class="token keyword">float</span> u<span class="token punctuation">,</span><span class="token keyword">float</span> v<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> <span class="token keyword">new</span> <span class="token class-name">Ray</span><span class="token punctuation">(</span>origin<span class="token punctuation">,</span> lower_left_corner <span class="token operator">+</span> u <span class="token operator">*</span> horizontal <span class="token operator">+</span> v <span class="token operator">*</span> vertical <span class="token operator">-</span> origin<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>最终的效果如下：<img src="Assets/Img/chapter6.png" alt="alt text" title="chapter6"><br>
放大看边缘的对比，还是很明显的。</p>
<h2 class="mume-header" id="%E7%AC%AC%E4%B8%83%E7%AB%A0-%E6%BC%AB%E5%8F%8D%E5%B0%84%E6%9D%90%E8%B4%A8diffuse-materials">第七章 漫反射材质（Diffuse Materials）</h2>
<p>我们可以渲染多个物体并且每个像素可以用多个射线采样了，现在我们来尝试一下渲染一些看起来真实一点的材质。我们就先从漫反射（亚光）材质开始吧。在实现中有一个问题：我们应该把材质和物体分开，让一个物体可以有不同的材质。或者把材质和物体绑定在一起，在一些几何体和材质绑定的过程纹理实现比较方便。本教程采用的是把材质和物体分开的方式，也是大多数渲染器采用的方式，但是也要注意这种方式的限制。<br>
漫反射，是投射在粗糙表面上的光向各个方向反射的现象。拥有这种特性的材质，表面粗糙，不发光，他们在光线的照射下会体现出本身固有的颜色。因为反射光线的方向是随机的，所以如下图，三束平行的光线照射反射的结果非常不同：<br>
<img src="Assets/figure/figure7-1.png" alt="alt text"><br>
到达漫反射的表面光线被吸收的要比被反射的多。因此颜色越深的物体，吸收的光线越多，这也是他们颜色深的原因。真随机方向算法，可以让物体表面看起来像磨砂的物体一样。<br>
下面我们来实现这种随机的算法。在射线与圆相交的地方，做一个单位圆（半径是1的）并与之前的圆相切，随机在单位圆上取一点S，这样S-P这个向量就可以看成是反射的方向。<br>
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mi>O</mi><mo>+</mo><mover accent="true"><mi>R</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">S = O + \vec R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span> 这里R就是x,y,z 在[-1,1]内随机在单位化一下（这里的实现跟原书不大一样，实测效果差不多）。所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>−</mo><mi>P</mi><mo>=</mo><mi>P</mi><mo>+</mo><mover accent="true"><mi>N</mi><mo>⃗</mo></mover><mo>+</mo><mover accent="true"><mi>R</mi><mo>⃗</mo></mover><mo>−</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">S- P = P + \vec N + \vec R - P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0496599999999998em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0496599999999998em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span></span></span></span> ，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>−</mo><mi>P</mi><mo>=</mo><mover accent="true"><mi>N</mi><mo>⃗</mo></mover><mo>+</mo><mover accent="true"><mi>R</mi><mo>⃗</mo></mover></mrow><annotation encoding="application/x-tex">S - P = \vec N + \vec R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0496599999999998em;vertical-align:-0.08333em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9663299999999999em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9663299999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span><span style="top:-3.25233em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.15216em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span></span></span></span> <img src="Assets/figure/figure7-2.png" alt="alt text"></p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">,</span> <span class="token class-name">Hitable</span> world<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Hit_record</span> rec<span class="token punctuation">;</span>
<span class="token keyword">float</span> min <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> max <span class="token operator">=</span> <span class="token keyword">float</span><span class="token punctuation">.</span>MaxValue<span class="token punctuation">;</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>world<span class="token punctuation">.</span><span class="token function">Hit</span><span class="token punctuation">(</span>ray<span class="token punctuation">,</span> <span class="token keyword">ref</span> min<span class="token punctuation">,</span> <span class="token keyword">ref</span> max<span class="token punctuation">,</span> <span class="token keyword">out</span> rec<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">var</span> target <span class="token operator">=</span> rec<span class="token punctuation">.</span>normal<span class="token punctuation">.</span>normalized <span class="token operator">+</span>
<span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span>Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">,</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">,</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">return</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token keyword">new</span> <span class="token class-name">Ray</span><span class="token punctuation">(</span>rec<span class="token punctuation">.</span>hitpoint<span class="token punctuation">,</span> target<span class="token punctuation">)</span><span class="token punctuation">,</span> world<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">else</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> unit_direction <span class="token operator">=</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">float</span> t <span class="token operator">=</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>unit_direction<span class="token punctuation">.</span>y <span class="token operator">+</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span><span class="token function">Lerp</span><span class="token punctuation">(</span>topColor<span class="token punctuation">,</span> bottomColor<span class="token punctuation">,</span> t<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>完整代码见chapter7.cs，给反射光线乘以0.5是假设光线有一半被吸收了。所以被光线反射次数越多的地方越暗，渲染结果如下图：<br>
<img src="Assets/Img/chapter7_1.png" alt="alt text" title="chapter7-1"><br>
下图是我们如果给图像做gamma矫正后的情况，gamma矫正是因为人眼对光强的变化不是线性的，根据gamma曲线做了矫正。<br>
<img src="Assets/Img/chapter7_2.png" alt="alt text" title="chapter7-2"></p>
<h2 class="mume-header" id="%E7%AC%AC%E5%85%AB%E7%AB%A0-%E9%87%91%E5%B1%9E%E6%9D%90%E8%B4%A8">第八章 金属材质</h2>
<p>如果我们想要物体用不同的材质，这里需要给物体一个接口，就是设置其材质的接口。为此抽象一个材质的接口：</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">interface</span> <span class="token class-name">IMaterial</span>
<span class="token punctuation">{</span>
<span class="token keyword">bool</span> <span class="token function">Scatter</span><span class="token punctuation">(</span><span class="token keyword">ref</span> <span class="token class-name">Ray</span> r<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Hit_record</span> rec<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Vector3</span> attenuation<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Ray</span> scattered<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>
</pre><p>光接触到物体表面，有一部分被吸收，有一部分被反射，这里的Scatter函数主要用来表述不同的材质，对光的作用，有多少吸收的，有多少是反射了，反射的方向又是怎样的。<br>
之前我们的Hit_record 也需要交点表面的材质信息</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">struct</span> Hit_record
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token keyword">float</span> t<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> hitpoint<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> normal<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token class-name">IMaterial</span> mat<span class="token punctuation">;</span> <span class="token comment">// 添加的材质信息</span>
<span class="token punctuation">}</span><span class="token punctuation">;</span>
</pre><p>把上一章的漫反射材质，封装一下写一个类：</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Lambertian</span> <span class="token punctuation">:</span> <span class="token class-name">IMaterial</span>
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> albedo<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">float</span> reflect<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token function">Lambertian</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> a<span class="token punctuation">,</span><span class="token keyword">float</span> r<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
albedo <span class="token operator">=</span> a<span class="token punctuation">;</span>
reflect <span class="token operator">=</span> r<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">bool</span> <span class="token function">Scatter</span><span class="token punctuation">(</span><span class="token keyword">ref</span> <span class="token class-name">Ray</span> r<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Hit_record</span> rec<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Vector3</span> attenuation<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Ray</span> scattered<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> target <span class="token operator">=</span> rec<span class="token punctuation">.</span>normal<span class="token punctuation">.</span>normalized <span class="token operator">+</span>
<span class="token keyword">new</span> <span class="token class-name">Vector3</span><span class="token punctuation">(</span>Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">,</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">,</span> Random<span class="token punctuation">.</span><span class="token function">Range</span><span class="token punctuation">(</span><span class="token operator">-</span><span class="token number">1f</span><span class="token punctuation">,</span> <span class="token number">1f</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
scattered<span class="token punctuation">.</span>origin <span class="token operator">=</span> rec<span class="token punctuation">.</span>hitpoint<span class="token punctuation">;</span>
scattered<span class="token punctuation">.</span>direction <span class="token operator">=</span> target<span class="token punctuation">;</span>
attenuation <span class="token operator">=</span> albedo <span class="token operator">*</span> reflect<span class="token punctuation">;</span>
<span class="token keyword">return</span> <span class="token keyword">true</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>这里比原书在对反射光的处理上多了一个系数reflect，就是假设有一部分光会被吸,反射的光线变暗了，这样效果更真实一点。</p>
<p>现在我们来讨论一下光滑的金属材质：光在遇到光滑的金属表面后几乎都被反射了，反射服从镜面反射定律，因此对金属材质来说，只要求的反射光线的方向就可以了<br>
原书的讲解不是很细致：这里放个链接感觉讲的要细一些：<a href="https://www.cnblogs.com/graphics/archive/2013/02/21/2920627.html">反射向量</a>。因为我们用了Unity中的Vector3这个类，里面有方法直接求的反射向量。这里就直接用了。</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"> <span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">MetalNoFuzz</span> <span class="token punctuation">:</span> <span class="token class-name">IMaterial</span>
<span class="token punctuation">{</span>
<span class="token keyword">public</span> <span class="token class-name">Vector3</span> albedo<span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token function">MetalNoFuzz</span><span class="token punctuation">(</span><span class="token class-name">Vector3</span> a<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
albedo <span class="token operator">=</span> a<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">public</span> <span class="token keyword">bool</span> <span class="token function">Scatter</span><span class="token punctuation">(</span><span class="token keyword">ref</span> <span class="token class-name">Ray</span> r<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Hit_record</span> rec<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Vector3</span> attenuation<span class="token punctuation">,</span> <span class="token keyword">ref</span> <span class="token class-name">Ray</span> scattered<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> reflected <span class="token operator">=</span> Vector3<span class="token punctuation">.</span><span class="token function">Reflect</span><span class="token punctuation">(</span>r<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">,</span> rec<span class="token punctuation">.</span>normal<span class="token punctuation">.</span>normalized<span class="token punctuation">)</span><span class="token punctuation">;</span>
scattered<span class="token punctuation">.</span>origin <span class="token operator">=</span> rec<span class="token punctuation">.</span>hitpoint<span class="token punctuation">;</span>
scattered<span class="token punctuation">.</span>direction <span class="token operator">=</span> reflected<span class="token punctuation">;</span>
attenuation <span class="token operator">=</span> albedo<span class="token punctuation">;</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span><span class="token function">Dot</span><span class="token punctuation">(</span>scattered<span class="token punctuation">.</span>direction<span class="token punctuation">,</span> rec<span class="token punctuation">.</span>normal<span class="token punctuation">)</span> <span class="token operator">></span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><p>最后我们在场景里面添加4个球，2个金属材质的，2个Lambertian材质的，做渲染：<br>
这里限定的光线最多反射50次</p>
<pre data-role="codeBlock" data-info="csharp" class="language-csharp"><span class="token keyword">private</span> <span class="token keyword">static</span> <span class="token keyword">int</span> MaxDepth <span class="token operator">=</span> <span class="token number">50</span><span class="token punctuation">;</span>
<span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token class-name">Vector3</span> <span class="token function">RayCast</span><span class="token punctuation">(</span><span class="token class-name">Ray</span> ray<span class="token punctuation">,</span> <span class="token class-name">Hitable</span> world<span class="token punctuation">,</span> <span class="token keyword">int</span> depth<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Hit_record</span> rec<span class="token punctuation">;</span>
<span class="token keyword">float</span> min <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
<span class="token keyword">float</span> max <span class="token operator">=</span> <span class="token keyword">float</span><span class="token punctuation">.</span>MaxValue<span class="token punctuation">;</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>world<span class="token punctuation">.</span><span class="token function">Hit</span><span class="token punctuation">(</span>ray<span class="token punctuation">,</span> <span class="token keyword">ref</span> min<span class="token punctuation">,</span> <span class="token keyword">ref</span> max<span class="token punctuation">,</span> <span class="token keyword">out</span> rec<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token class-name">Ray</span> scattered <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Ray</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token class-name">Vector3</span> attenuation <span class="token operator">=</span> Vector3<span class="token punctuation">.</span>one<span class="token punctuation">;</span>
<span class="token keyword">if</span> <span class="token punctuation">(</span>depth <span class="token operator"><</span> MaxDepth <span class="token operator">&&</span> rec<span class="token punctuation">.</span>mat<span class="token punctuation">.</span><span class="token function">Scatter</span><span class="token punctuation">(</span><span class="token keyword">ref</span> ray<span class="token punctuation">,</span> <span class="token keyword">ref</span> rec<span class="token punctuation">,</span> <span class="token keyword">ref</span> attenuation<span class="token punctuation">,</span> <span class="token keyword">ref</span> scattered<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">var</span> color <span class="token operator">=</span> <span class="token function">RayCast</span><span class="token punctuation">(</span>scattered<span class="token punctuation">,</span> world<span class="token punctuation">,</span> depth <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
attenuation<span class="token punctuation">.</span>x <span class="token operator">*=</span> color<span class="token punctuation">.</span>x<span class="token punctuation">;</span>
attenuation<span class="token punctuation">.</span>y <span class="token operator">*=</span> color<span class="token punctuation">.</span>y<span class="token punctuation">;</span>
attenuation<span class="token punctuation">.</span>z <span class="token operator">*=</span> color<span class="token punctuation">.</span>z<span class="token punctuation">;</span>
<span class="token keyword">return</span> attenuation<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">else</span>
<span class="token punctuation">{</span>
<span class="token keyword">return</span> Vector3<span class="token punctuation">.</span>zero<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">else</span>
<span class="token punctuation">{</span>
<span class="token class-name">Vector3</span> unit_direction <span class="token operator">=</span> ray<span class="token punctuation">.</span>direction<span class="token punctuation">.</span>normalized<span class="token punctuation">;</span>
<span class="token keyword">float</span> t <span class="token operator">=</span> <span class="token number">0.5f</span> <span class="token operator">*</span> <span class="token punctuation">(</span>unit_direction<span class="token punctuation">.</span>y <span class="token operator">+</span> <span class="token number">1.0f</span><span class="token punctuation">)</span><span class="token punctuation">;</span>