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<h1 class="indigo-link-context"><a href="/ifs-demystified/">IFS 分形揭秘</a></h1>
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<time class="pink-link-context" datetime="2024-06-16T00:00:00.000Z"><a href="/ifs-demystified/">2024-06-16</a></time>
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<p>本文整理自我 2024 年 6 月 14
日在上海科技大学数学所的一个小报告，标题是「GPU
涂鸦与数学可视化」。我保留了报告的技术内容，略去了报告中介绍 demoscene
文化和分形文化的部分。</p>
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<h1 class="indigo-link-context"><a href="/envelope-and-caustics/">咖啡杯中的焦散线</a></h1>
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<time class="pink-link-context" datetime="2024-02-08T16:00:00.000Z"><a href="/envelope-and-caustics/">2024-02-09</a></time>
<div class="entry pink-link-context">
<p>春节的晚上，外面鞭炮喧天，家人在看电视，我躲在屋里看数学，还是挺惬意的。</p>
<p>我看的是 <a target="_blank" rel="noopener" href="https://johncarlosbaez.wordpress.com">John Baez</a>
和 <a target="_blank" rel="noopener" href="https://www.gregegan.net">Greg Egan</a> 的博客。John Baez
是一位在科普方面非常高产的数学家，写过不计其数的科普文章。读他的文章非常让人享受，因为他总是从直观的例子入手，一步步启发读者，展开到更高级的数学。Greg
Egan
是澳大利亚的一位非常高产的科幻小说作家，有不少作品已经被国内引入。他的小说属于硬科幻风格，而且是非常硬的那种。他也有不少有趣的
<a target="_blank" rel="noopener" href="https://www.gregegan.net/SCIENCE/Science.html">博客文章</a>。不过与
John Baez 不同的是，Greg Egan
的文章不太会去兼顾不同水平的读者，对我来说，要看懂他在说什么经常不是一件容易的事情。</p>
<p>John Baez 博客上有一个系列 <a target="_blank" rel="noopener" href="https://johncarlosbaez.wordpress.com/2012/08/31/rolling-circles-and-balls-part-1/">Rolling
circles and balls</a> 讨论了圆的外摆线和焦散，Greg Egan 也有一篇 <a target="_blank" rel="noopener" href="https://www.gregegan.net/SCIENCE/Catacaustics/Catacaustics.html">文章</a>
更深入的讨论了曲线的焦散。这个话题非常有意思，我也一时手痒写代码实验了一番并记录在此。</p>
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<h1 class="indigo-link-context"><a href="/mobius-ball-rigid-motion/">Möbius 变换与球的刚体运动</a></h1>
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<time class="pink-link-context" datetime="2022-05-07T16:00:00.000Z"><a href="/mobius-ball-rigid-motion/">2022-05-08</a></time>
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<p>五一期间我写了一个 shadertoy 小动画，演示 Möbius
变换与球的刚体运动之间的关系：</p>
</div>
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<a href="/mobius-ball-rigid-motion/">阅读全文</a>
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<h1 class="indigo-link-context"><a href="/mobius-poincare/">Möbius 变换与二维 Poincaré 双曲空间的等距</a></h1>
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<time class="pink-link-context" datetime="2022-03-21T16:00:00.000Z"><a href="/mobius-poincare/">2022-03-22</a></time>
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<p>这几天因为疫情居家观察难得多出点时间（不用健步挤地铁），可以写点小文章。我之前写过一篇介绍
<a href="/mobius-h3space">Möbius
变换分类的文章</a>，今天继续那里的讨论，介绍一个 Möbius 变换 <span class="math inline">\(M\)</span> 作为二维 Poincaré 双曲圆盘 <span class="math inline">\(\mathbb{D}\)</span> 中等距的两种构造方法：</p>
<ol type="1">
<li>指定 <span class="math inline">\(M\)</span>
的不动点的个数和位置，不动点的个数和位置可以决定变换的类型。</li>
<li>指定两个反射镜面的位置：取两条测地线作为镜面，则关于这两个镜面的反演变换的复合变换就是一个
Möbius 变换。两个镜面的相对位置可以决定变换的类型。</li>
</ol>
<p>这两种方法分别对应在 <span class="math inline">\(M\)</span>
作用下保持不变的两个圆族。</p>
<p>本文的插图使用 <a target="_blank" rel="noopener" href="https://matplotlib.org/">matplotlib</a>
绘制，代码在 <a target="_blank" rel="noopener" href="https://github.com/neozhaoliang/pywonderland/tree/master/src/mobius">Github</a>
上。</p>
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<a href="/mobius-poincare/">阅读全文</a>
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<div class="article-title">
<h1 class="indigo-link-context"><a href="/Rouche-theorem-winding-number/">遛狗中的数学：曲线的环绕数、Rouché 定理和开映射定理</a></h1>
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<time class="pink-link-context" datetime="2021-08-31T16:00:00.000Z"><a href="/Rouche-theorem-winding-number/">2021-09-01</a></time>
<div class="entry pink-link-context">
<p>我写了一个 <a target="_blank" rel="noopener" href="https://www.shadertoy.com/view/fdK3RD">shadertoy
小动画</a>，演示 <span class="citation" data-cites="Needham1997">(<a href="#ref-Needham1997" role="doc-biblioref">Needham 1997</a>)</span>
书中第 7 章 “Winding numbers and topology” 中的结论：</p>
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