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import{_ as n,c,a as l,b as s,d as a,o as t}from"./app-BdNP3b8G.js";const x="/images/2015/05/rnd_01.gif",e="/images/2015/05/rnd_02.gif",j="/images/2015/05/rnd_03.gif",i="/images/2015/05/rnd_04.png",o="/images/2015/05/rnd_07.png",p={};function r(u,m){return t(),c("div",null,m[0]||(m[0]=[l(`<h1 id="如何生成一个随机的圆形" tabindex="-1"><a class="header-anchor" href="#如何生成一个随机的圆形"><span>如何生成一个随机的圆形</span></a></h1><p>最近在工作中遇到这么一个问题：</p><blockquote><p>在游戏场景中有一个怪物生成点，这个生长点产生的怪物均匀分布在半径为R的圆形内，这个随机算法应该如何生成？看起来很简单，随手写了一个：</p></blockquote><div class="language-cpp" data-highlighter="prismjs" data-ext="cpp" data-title="cpp"><pre><code><span class="line"><span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">define</span> <span class="token macro-name">RAND</span> <span class="token expression"><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token function">rand</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token operator">/</span>RAND_MAX<span class="token punctuation">)</span></span></span></span>
<span class="line"> </span>
<span class="line"><span class="token keyword">void</span> <span class="token function">get_random_pos</span><span class="token punctuation">(</span><span class="token keyword">float</span> center_x<span class="token punctuation">,</span> <span class="token keyword">float</span> center_y<span class="token punctuation">,</span> <span class="token keyword">float</span> radius<span class="token punctuation">,</span> <span class="token keyword">float</span><span class="token operator">&amp;</span>x<span class="token punctuation">,</span> <span class="token keyword">float</span><span class="token operator">&amp;</span> y<span class="token punctuation">)</span></span>
<span class="line"><span class="token punctuation">{</span></span>
<span class="line"> <span class="token keyword">float</span> u <span class="token operator">=</span> RAND<span class="token operator">*</span>radius<span class="token punctuation">;</span></span>
<span class="line"> <span class="token keyword">float</span> v <span class="token operator">=</span> RAND<span class="token operator">*</span><span class="token number">2</span><span class="token operator">*</span>PI<span class="token punctuation">;</span></span>
<span class="line"> </span>
<span class="line"> x <span class="token operator">=</span> center_x <span class="token operator">+</span> u<span class="token operator">*</span><span class="token function">cos</span><span class="token punctuation">(</span>v<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line"> y <span class="token operator">=</span> center_y <span class="token operator">+</span> u<span class="token operator">*</span><span class="token function">sin</span><span class="token punctuation">(</span>v<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line"><span class="token punctuation">}</span></span>
<span class="line"></span></code></pre></div><p>但写的过程中，直觉告诉我，这么写肯定是有问题的，试想，如果以北京为例，如果北京的人口是均匀分布的，那么如果让所有人都报出自己家和天安门的距离，那么这些数据肯定不是均匀，因为居住在五环附近的人数肯定要大于居住在二环附近的人数，于是用Mathamatica实验一下：</p><figure><img src="`+x+'" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>果然，这么写是不对的，网上查了一下，这个问题还真是有人研究过，说应该把所获得的随机数开平方一下，实验一下：</p><figure><img src="'+e+'" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure>',8),s("p",null,[a("但是，这个开平方背后的数学原理究竟是什么呢？抽空翻了下概率书，原来，其中的道理并不复杂，这里涉及到概率里的一个基本概念，累计分布函数（Cumulative distribution function），简称CFD，它的定义如下： 设有一个随机变量"),s("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[s("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[s("mjx-mi",{class:"mjx-i"},[s("mjx-c",{class:"mjx-c1D44B 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TEX-I"})]),s("mjx-mo",{class:"mjx-n"},[s("mjx-c",{class:"mjx-c29"})]),s("mjx-mo",{class:"mjx-n",space:"4"},[s("mjx-c",{class:"mjx-c3D"})]),s("mjx-mi",{class:"mjx-i",space:"4"},[s("mjx-c",{class:"mjx-c1D445 TEX-I"})]),s("mjx-msqrt",null,[s("mjx-sqrt",null,[s("mjx-surd",null,[s("mjx-mo",{class:"mjx-n"},[s("mjx-c",{class:"mjx-c221A"})])]),s("mjx-box",{style:{"padding-top":"0.329em"}},[s("mjx-mi",{class:"mjx-i"},[s("mjx-c",{class:"mjx-c1D462 TEX-I"})])])])])]),s("mjx-assistive-mml",{unselectable:"on",display:"block"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("msup",null,[s("mi",null,"F"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"−"),s("mn",null,"1")])]),s("mo",{stretchy:"false"},"("),s("mi",null,"u"),s("mo",{stretchy:"false"},")"),s("mo",null,"="),s("mi",null,"R"),s("msqrt",null,[s("mi",null,"u")])])])],-1),l(`<p>所以最终的算法可以写成</p><div class="language-cpp" data-highlighter="prismjs" data-ext="cpp" data-title="cpp"><pre><code><span class="line"><span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">define</span> <span class="token macro-name">RAND</span> <span class="token expression"><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token keyword">float</span><span class="token punctuation">)</span><span class="token function">rand</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token operator">/</span>RAND_MAX<span class="token punctuation">)</span></span></span></span>
<span class="line"> </span>
<span class="line"><span class="token keyword">void</span> <span class="token function">get_random_pos</span><span class="token punctuation">(</span><span class="token keyword">float</span> center_x<span class="token punctuation">,</span> <span class="token keyword">float</span> center_y<span class="token punctuation">,</span> <span class="token keyword">float</span> radius<span class="token punctuation">,</span> <span class="token keyword">float</span><span class="token operator">&amp;</span>x<span class="token punctuation">,</span> <span class="token keyword">float</span><span class="token operator">&amp;</span> y<span class="token punctuation">)</span></span>
<span class="line"><span class="token punctuation">{</span></span>
<span class="line"> <span class="token keyword">float</span> u <span class="token operator">=</span> <span class="token function">sqrt</span><span class="token punctuation">(</span>RAND<span class="token punctuation">)</span><span class="token operator">*</span>radius<span class="token punctuation">;</span></span>
<span class="line"> <span class="token keyword">float</span> v <span class="token operator">=</span> RAND<span class="token operator">*</span><span class="token number">2</span><span class="token operator">*</span>PI<span class="token punctuation">;</span></span>
<span class="line"> </span>
<span class="line"> x <span class="token operator">=</span> center_x <span class="token operator">+</span> u<span class="token operator">*</span><span class="token function">cos</span><span class="token punctuation">(</span>v<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line"> y <span class="token operator">=</span> center_y <span class="token operator">+</span> u<span class="token operator">*</span><span class="token function">sin</span><span class="token punctuation">(</span>v<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line"><span class="token punctuation">}</span></span>
<span class="line"></span></code></pre></div>`,2)]))}const h=n(p,[["render",r],["__file","RandRound.html.vue"]]),y=JSON.parse('{"path":"/blog/2025/02/RandRound.html","title":"如何生成一个随机的圆形","lang":"zh-CN","frontmatter":{"title":"如何生成一个随机的圆形","tags":"数学 概率 程序 算法"},"headers":[],"git":{"updatedTime":1740310507000,"contributors":[{"name":"thejinchao","username":"thejinchao","email":"thejinchao@gmail.com","commits":2,"url":"https://github.com/thejinchao"}]},"filePathRelative":"blog/2025/02/RandRound.md"}');export{h as comp,y as data};
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