677 lines
92 KiB
HTML
677 lines
92 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="zh-CN">
|
||
<head prefix="og: http://ogp.me/ns# article: http://ogp.me/ns/article#">
|
||
<meta charset="UTF-8" />
|
||
|
||
<meta name="generator" content="Hugo 0.113.0"><meta name="theme-color" content="#fff" />
|
||
<meta name="color-scheme" content="light dark">
|
||
|
||
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
||
|
||
<meta name="format-detection" content="telephone=no, date=no, address=no, email=no" />
|
||
|
||
<meta http-equiv="Cache-Control" content="no-transform" />
|
||
|
||
<meta http-equiv="Cache-Control" content="no-siteapp" />
|
||
|
||
<title>几何（geometry） | 飞鸿踏雪</title>
|
||
|
||
<link rel="stylesheet" href="/css/meme.min.12a3541920a798d49c1bca858f15c53090f14ce9367a34faa2bb51dd38299730.css"/>
|
||
|
||
|
||
|
||
<script src="/js/meme.min.ac2b48f322cff6a9a8036be3baf3e59e300cd9e4191997a8c3765f79de02bf8d.js"></script>
|
||
|
||
|
||
|
||
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
|
||
|
||
<link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=EB+Garamond:ital,wght@0,400;0,500;0,700;1,400;1,700&family=Noto+Serif+SC:wght@400;500;700&family=Source+Code+Pro:ital,wght@0,400;0,700;1,400;1,700&display=swap" media="print" onload="this.media='all'" />
|
||
<noscript><link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=EB+Garamond:ital,wght@0,400;0,500;0,700;1,400;1,700&family=Noto+Serif+SC:wght@400;500;700&family=Source+Code+Pro:ital,wght@0,400;0,700;1,400;1,700&display=swap" /></noscript>
|
||
|
||
<meta name="author" content="InkSoul" /><meta name="description" content="几何 几何物体的表现形式 隐式(Implicit) 代数平面(algebraic surface) 水平集(……" />
|
||
|
||
<link rel="shortcut icon" href="/favicon.ico" type="image/x-icon" />
|
||
<link rel="mask-icon" href="/icons/safari-pinned-tab.svg" color="#2a6df4" />
|
||
<link rel="apple-touch-icon" sizes="180x180" href="/icons/apple-touch-icon.png" />
|
||
<meta name="apple-mobile-web-app-capable" content="yes" />
|
||
<meta name="apple-mobile-web-app-title" content="飞鸿踏雪" />
|
||
<meta name="apple-mobile-web-app-status-bar-style" content="black" />
|
||
<meta name="mobile-web-app-capable" content="yes" />
|
||
<meta name="application-name" content="飞鸿踏雪" />
|
||
<meta name="msapplication-starturl" content="../../" />
|
||
<meta name="msapplication-TileColor" content="#fff" />
|
||
<meta name="msapplication-TileImage" content="../../icons/mstile-150x150.png" />
|
||
<link rel="manifest" href="/manifest.json" />
|
||
|
||
|
||
|
||
|
||
|
||
<link rel="canonical" href="http://www.inksoul.top/computergraphic/%E5%87%A0%E4%BD%95geometry/" />
|
||
|
||
|
||
<script type="application/ld+json">
|
||
{
|
||
"@context": "https://schema.org",
|
||
"@type": "BlogPosting",
|
||
"datePublished": "2022-07-15T14:40:43+08:00",
|
||
"dateModified": "2023-06-18T21:19:01+08:00",
|
||
"url": "http://www.inksoul.top/computergraphic/%E5%87%A0%E4%BD%95geometry/",
|
||
"headline": "几何（geometry）",
|
||
"description": "几何 几何物体的表现形式 隐式(Implicit) 代数平面(algebraic surface) 水平集(……",
|
||
"inLanguage" : "zh-CN",
|
||
"articleSection": "computergraphic",
|
||
"wordCount": 2777 ,
|
||
"image": ["http://www.inksoul.top/../../images/eg_sample_can_be_hard.png","http://www.inksoul.top/../../images/eg_in_outside_test_easy.png","http://www.inksoul.top/../../images/Algebraic_Surfaces.png","http://www.inksoul.top/../../images/Constructive_Solid_Geometry.png","http://www.inksoul.top/../../images/eg_Distance_Functions.png","http://www.inksoul.top/../../images/Level_Set_grid.png","http://www.inksoul.top/../../images/eg_Factals.png","http://www.inksoul.top/../../images/eg_explict_mapping.png","http://www.inksoul.top/../../images/eg_point_cloud.png","http://www.inksoul.top/../../images/eg_polygon_mesh.png","http://www.inksoul.top/../../images/David_Baraff.png","http://www.inksoul.top/../../images/Defining_Bezier_Curve_Tangents.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step1.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step2.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step3.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step4.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step5.png","http://www.inksoul.top/../../images/de_Casteljau_Algorithm_step6.png","http://www.inksoul.top/../../images/Visualizing_de_Casteljau.png","http://www.inksoul.top/../../images/de_Casteljau_pyramid_of.png","http://www.inksoul.top/../../images/Cubic_Bezier_Basis_Functions.png","http://www.inksoul.top/../../images/Demon_Piecewise_Cubic_Bezier_Curve.png","http://www.inksoul.top/../../images/Continuity_Piecewise_Bezier_step1.png","http://www.inksoul.top/../../images/Continuity_Piecewise_Bezier_step2.png","http://www.inksoul.top/../../images/Continuity_Piecewise_Bezier_step3.png","http://www.inksoul.top/../../images/bezier_surface_eg.png","http://www.inksoul.top/../../images/evaluation_of_bezier_surface.png","http://www.inksoul.top/../../images/visual_evaluation_of_bezier_surface.png","http://www.inksoul.top/../../images/eg_mesh_subdivision.png","http://www.inksoul.top/../../images/loop_subdivision_new.png","http://www.inksoul.top/../../images/loop_subdivision_old.png","http://www.inksoul.top/../../images/catmull-clark_subdivision1.png","http://www.inksoul.top/../../images/catmull-clark_subdivision2.png","http://www.inksoul.top/../../images/Catmull-Clark_Vertex_Update_Rules.png","http://www.inksoul.top/../../images/convergence_of_loop_and_catmull.png","http://www.inksoul.top/../../images/eg_mesh_simplification.png","http://www.inksoul.top/../../images/Collapsing_An_Edge.png","http://www.inksoul.top/../../images/Quadric_Error_Metrics.png","http://www.inksoul.top/../../images/eg_Quadric_Error_Mesh_Simplification.png","http://www.inksoul.top/../../images/render_from_light.png","http://www.inksoul.top/../../images/project_to_light.png"],
|
||
"author": {
|
||
"@type": "Person",
|
||
"email": "qingci30@163.com",
|
||
"image": "http://www.inksoul.top/icons/apple-touch-icon.png",
|
||
"url": "http://www.inksoul.top",
|
||
"name": "InkSoul"
|
||
},
|
||
"license": "[CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh)",
|
||
"publisher": {
|
||
"@type": "Organization",
|
||
"name": "飞鸿踏雪",
|
||
"logo": {
|
||
"@type": "ImageObject",
|
||
"url": "http://www.inksoul.top/icons/apple-touch-icon.png"
|
||
},
|
||
"url": "http://www.inksoul.top/"
|
||
},
|
||
"mainEntityOfPage": {
|
||
"@type": "WebSite",
|
||
"@id": "http://www.inksoul.top/"
|
||
}
|
||
}
|
||
</script>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<meta property="og:title" content="几何（geometry）" />
|
||
<meta property="og:description" content="几何 几何物体的表现形式 隐式(Implicit) 代数平面(algebraic surface) 水平集(……" />
|
||
<meta property="og:url" content="http://www.inksoul.top/computergraphic/%E5%87%A0%E4%BD%95geometry/" />
|
||
<meta property="og:site_name" content="飞鸿踏雪" />
|
||
<meta property="og:locale" content="zh" /><meta property="og:image" content="http://www.inksoul.top/../../images/eg_sample_can_be_hard.png" />
|
||
<meta property="og:type" content="article" />
|
||
<meta property="article:published_time" content="2022-07-15T14:40:43+08:00" />
|
||
<meta property="article:modified_time" content="2023-06-18T21:19:01+08:00" />
|
||
|
||
<meta property="article:section" content="computergraphic" />
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</head>
|
||
|
||
<body>
|
||
<div class="container">
|
||
|
||
<header class="header">
|
||
|
||
<div class="header-wrapper">
|
||
<div class="header-inner single">
|
||
|
||
<div class="site-brand">
|
||
|
||
<a href="/" class="brand">飞鸿踏雪</a>
|
||
|
||
</div>
|
||
|
||
<nav class="nav">
|
||
<ul class="menu" id="menu">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="menu-item active"><a href="/computergraphic/"><svg class="icon computergraphic" fill="none" stroke="currentColor" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M20 7l-8-4-8 4m16 0l-8 4m8-4v10l-8 4m0-10L4 7m8 4v10M4 7v10l8 4"></path></svg><span class="menu-item-name">图形学</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
<li class="menu-item"><a href="/algorithm/"><svg class="icon algorithm" fill="none" stroke="currentColor" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M9.663 17h4.673M12 3v1m6.364 1.636l-.707.707M21 12h-1M4 12H3m3.343-5.657l-.707-.707m2.828 9.9a5 5 0 117.072 0l-.548.547A3.374 3.374 0 0014 18.469V19a2 2 0 11-4 0v-.531c0-.895-.356-1.754-.988-2.386l-.548-.547z"></path></svg><span class="menu-item-name">算法</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
<li class="menu-item"><a href="/mathematics/"><svg class="icon mathematics" fill="none" stroke="currentColor" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 576 512"><path stroker-linecap="round" stroke-linejoin="round" stroker-width="2" d="M576 32.01c0-17.69-14.33-31.1-32-31.1l-224-.0049c-14.69 0-27.48 10-31.05 24.25L197.9 388.3L124.6 241.7C119.2 230.9 108.1 224 96 224L32 224c-17.67 0-32 14.31-32 31.1s14.33 32 32 32h44.22l103.2 206.3c5.469 10.91 16.6 17.68 28.61 17.68c1.172 0 2.323-.0576 3.495-.1826c13.31-1.469 24.31-11.06 27.56-24.06l105.9-423.8H544C561.7 64.01 576 49.7 576 32.01zM566.6 233.4c-12.5-12.5-32.75-12.5-45.25 0L480 274.8l-41.38-41.37c-12.5-12.5-32.75-12.5-45.25 0s-12.5 32.75 0 45.25l41.38 41.38l-41.38 41.38c-12.5 12.5-12.5 32.75 0 45.25C399.6 412.9 407.8 416 416 416s16.38-3.125 22.62-9.375L480 365.3l41.38 41.38C527.6 412.9 535.8 416 544 416s16.38-3.125 22.62-9.375c12.5-12.5 12.5-32.75 0-45.25l-41.38-41.38L566.6 278.6C579.1 266.1 579.1 245.9 566.6 233.4z"/></svg><span class="menu-item-name">数学</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
<li class="menu-item"><a href="/408/"><svg class="icon 408" fill="none" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 576 512"><!--! Font Awesome Pro 6.1.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license (Commercial License) Copyright 2022 Fonticon 408s, Inc. --><path d="M172.1 40.16L268.1 3.76C280.9-1.089 295.1-1.089 307.9 3.76L403.9 40.16C425.6 48.41 440 69.25 440 92.52V204.7C441.3 205.1 442.6 205.5 443.9 205.1L539.9 242.4C561.6 250.6 576 271.5 576 294.7V413.9C576 436.1 562.9 456.2 542.5 465.1L446.5 507.3C432.2 513.7 415.8 513.7 401.5 507.3L288 457.5L174.5 507.3C160.2 513.7 143.8 513.7 129.5 507.3L33.46 465.1C13.13 456.2 0 436.1 0 413.9V294.7C0 271.5 14.39 250.6 36.15 242.4L132.1 205.1C133.4 205.5 134.7 205.1 136 204.7V92.52C136 69.25 150.4 48.41 172.1 40.16V40.16zM290.8 48.64C289 47.95 286.1 47.95 285.2 48.64L206.8 78.35L287.1 109.5L369.2 78.35L290.8 48.64zM392 210.6V121L309.6 152.6V241.8L392 210.6zM154.8 250.9C153 250.2 150.1 250.2 149.2 250.9L70.81 280.6L152 311.7L233.2 280.6L154.8 250.9zM173.6 455.3L256 419.1V323.2L173.6 354.8V455.3zM342.8 280.6L424 311.7L505.2 280.6L426.8 250.9C425 250.2 422.1 250.2 421.2 250.9L342.8 280.6zM528 413.9V323.2L445.6 354.8V455.3L523.2 421.2C526.1 419.9 528 417.1 528 413.9V413.9z"/></svg><span class="menu-item-name">408</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
<li class="menu-item"><a href="/stuff/"><svg class="icon stuff" fill="none" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Pro 6.1.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license (Commercial License) Copyright 2022 Fonticon stuffs, Inc. --><path d="M216 104C202.8 104 192 114.8 192 128s10.75 24 24 24c79.41 0 144 64.59 144 144C360 309.3 370.8 320 384 320s24-10.75 24-24C408 190.1 321.9 104 216 104zM224 0C206.3 0 192 14.31 192 32s14.33 32 32 32c123.5 0 224 100.5 224 224c0 17.69 14.33 32 32 32s32-14.31 32-32C512 129.2 382.8 0 224 0zM188.9 346l27.37-27.37c2.625 .625 5.059 1.506 7.809 1.506c17.75 0 31.99-14.26 31.99-32c0-17.62-14.24-32.01-31.99-32.01c-17.62 0-31.99 14.38-31.99 32.01c0 2.875 .8099 5.25 1.56 7.875L166.2 323.4L49.37 206.5c-7.25-7.25-20.12-6-24.1 3c-41.75 77.88-29.88 176.7 35.75 242.4c65.62 65.62 164.6 77.5 242.4 35.75c9.125-5 10.38-17.75 3-25L188.9 346z"/></svg><span class="menu-item-name">杂物</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
<li class="menu-item"><a href="/about/"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512" class="icon user-circle"><path d="M248 8C111 8 0 119 0 256s111 248 248 248 248-111 248-248S385 8 248 8zm0 96c48.6 0 88 39.4 88 88s-39.4 88-88 88-88-39.4-88-88 39.4-88 88-88zm0 344c-58.7 0-111.3-26.6-146.5-68.2 18.8-35.4 55.6-59.8 98.5-59.8 2.4 0 4.8.4 7.1 1.1 13 4.2 26.6 6.9 40.9 6.9 14.3 0 28-2.7 40.9-6.9 2.3-.7 4.7-1.1 7.1-1.1 42.9 0 79.7 24.4 98.5 59.8C359.3 421.4 306.7 448 248 448z"/></svg><span class="menu-item-name">关于</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="menu-item">
|
||
<a id="theme-switcher" href="#"><span class="icon theme-icon-light">🌞</span><span class="icon theme-icon-dark">🌙</span></a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
|
||
<input type="checkbox" id="nav-toggle" aria-hidden="true" />
|
||
<label for="nav-toggle" class="nav-toggle"></label>
|
||
<label for="nav-toggle" class="nav-curtain"></label>
|
||
|
||
|
||
|
||
</header>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<main class="main single" id="main">
|
||
<div class="main-inner">
|
||
|
||
|
||
|
||
<article class="content post h-entry" data-align="justify" data-type="computergraphic" data-toc-num="true">
|
||
|
||
<h1 class="post-title p-name">几何（geometry）</h1>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="post-body e-content">
|
||
<h3 id="几何"><a href="#几何" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>几何</h3>
|
||
<h4 id="几何物体的表现形式"><a href="#几何物体的表现形式" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>几何物体的表现形式</h4>
|
||
<h5 id="隐式implicit"><a href="#隐式implicit" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>隐式(Implicit)</h5>
|
||
<ol>
|
||
<li>代数平面(algebraic surface)</li>
|
||
<li>水平集(level sets)</li>
|
||
<li>距离函数(distance functions)</li>
|
||
<li>...</li>
|
||
</ol>
|
||
<h6 id="优缺点"><a href="#优缺点" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>优缺点</h6>
|
||
<p>优点：</p>
|
||
<ol>
|
||
<li>描述简洁(如，一个函数)</li>
|
||
<li>便于某些查询(判定物体内部，或内部点到表面的距离)</li>
|
||
<li>便于计算光线到表面的交集</li>
|
||
<li>对于简单的形状能够做到准确描述无抽样误差</li>
|
||
<li>易于处理拓扑变化(如，流体)</li>
|
||
</ol>
|
||
<p>缺点：</p>
|
||
<ol>
|
||
<li>难以模拟复杂的形状</li>
|
||
</ol>
|
||
<h6 id="基于一系列满足特定关系的点"><a href="#基于一系列满足特定关系的点" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>基于一系列满足特定关系的点</h6>
|
||
<p>例如：</p>
|
||
<p>球体：所有的点在三维坐标系里满足$$x^2 + y^2 + z^2 = 1$$或$$f(x,y,z) = 0 $$</p>
|
||
<h6 id="难以采样"><a href="#难以采样" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>难以采样</h6>
|
||
<p>例如：</p>
|
||
<p>在一系列符合下列条件的点中，难以采样到符合$f(x,y,z) = 0$的点</p>
|
||
<p>$$f(x,y,z) = (2 - \sqrt{x^2 + y^2})^2 + z^2 - 1$$</p>
|
||
<p>对应的几何形状如下图</p>
|
||
<p><img src="../../images/eg_sample_can_be_hard.png" alt=""></p>
|
||
<h6 id="易于判断内外关系"><a href="#易于判断内外关系" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>易于判断内外关系</h6>
|
||
<p>例如：</p>
|
||
<p>在一系列符合下述条件的点中</p>
|
||
<p>$$f(x,y,z) = x^2 + y^2 + z^2 - 1$$</p>
|
||
<p><img src="../../images/eg_in_outside_test_easy.png" alt=""></p>
|
||
<p>对于点$(\frac{3}{4},\frac{1}{2},\frac{1}{4})$ 若要判定其是否在该几何体内部则只需计算$$f(x,y,z) = - \frac{1}{8} < 0$$ 即可判定其位于几何体内部</p>
|
||
<hr>
|
||
<h6 id="代数平面algebraic-surfaces"><a href="#代数平面algebraic-surfaces" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>代数平面(Algebraic Surfaces)</h6>
|
||
<p>表面是x,y,z中多项式的零集</p>
|
||
<p><img src="../../images/Algebraic_Surfaces.png" alt=""></p>
|
||
<h6 id="构造实体几何constructive-solid-geometry"><a href="#构造实体几何constructive-solid-geometry" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>构造实体几何(Constructive Solid Geometry)</h6>
|
||
<p>通过布尔计算组合构造隐式几何</p>
|
||
<p><img src="../../images/Constructive_Solid_Geometry.png" alt=""></p>
|
||
<h6 id="距离函数distance-functions"><a href="#距离函数distance-functions" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>距离函数(Distance Functions)</h6>
|
||
<p>距离函数：从任意位置到目标物体给出最小距离（符号距离）</p>
|
||
<p>使用距离函数将两个曲面混合在一起</p>
|
||
<p><img src="../../images/eg_Distance_Functions.png" alt=""></p>
|
||
<h6 id="水平集level-set-method"><a href="#水平集level-set-method" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>水平集(Level Set Method)</h6>
|
||
<p>封闭式的方程难以描述复杂的形状</p>
|
||
<p>解决方案：存储值相似的函数网格</p>
|
||
<p><img src="../../images/Level_Set_grid.png" alt=""></p>
|
||
<p>插值为零的值的位置即为表面</p>
|
||
<p>优势：能够提供对形状更明确的控制（如纹理）</p>
|
||
<p>在流体仿真中也存在应用：计算到气液边界的距离</p>
|
||
<h6 id="分形factals"><a href="#分形factals" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>分形(Factals)</h6>
|
||
<p>该几何形状表现为所有尺度的细节都存在自相似性(一种描述自然现象的说法)，往往难以控制形状</p>
|
||
<p><img src="../../images/eg_Factals.png" alt=""></p>
|
||
<hr>
|
||
<h5 id="显式explicit"><a href="#显式explicit" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>显式(Explicit)</h5>
|
||
<ol>
|
||
<li>点云(point cloud)</li>
|
||
<li>多边形网格(polygon mesh)</li>
|
||
<li>细分曲面和曲线(subdivision, NURBS)</li>
|
||
<li>...</li>
|
||
</ol>
|
||
<h6 id="点直接或参数映射给出"><a href="#点直接或参数映射给出" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>点直接或参数映射给出</h6>
|
||
<p>例如：</p>
|
||
<p>$$f:R^2 \rightarrow R^3;(u,v) \rightarrow (x,y,z)$$</p>
|
||
<p><img src="../../images/eg_explict_mapping.png" alt=""></p>
|
||
<h6 id="易于采样"><a href="#易于采样" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>易于采样</h6>
|
||
<p>例如：</p>
|
||
<p>对于$$ f(u,v) = ((2 + \cos u)\cos v,(2 + \cos u)\sin v,\sin u) $$</p>
|
||
<p>若要判定点$f(u,v)$是否位于表面，则只需将$(u,v)$的值相加</p>
|
||
<p><img src="../../images/eg_sample_can_be_hard.png" alt=""></p>
|
||
<h6 id="难以判断内外关系"><a href="#难以判断内外关系" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>难以判断内外关系</h6>
|
||
<p>例如：
|
||
对于$$f(u,v) = (\cos u \sin v ,\sin u \sin v,\cos v)$$</p>
|
||
<p>难以判定点$(\frac{3}{4},\frac{1}{2},\frac{1}{4})$</p>
|
||
<hr>
|
||
<h6 id="点云-point-cloud"><a href="#点云-point-cloud" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>点云 (Point Cloud)</h6>
|
||
<ol>
|
||
<li>最简单的表示：点列表(x,y,z)</li>
|
||
<li>轻松表现任何类型的几何图形</li>
|
||
<li>适用于大型数据集(>> 1 点/像素)</li>
|
||
<li>通常转换为多边形网格</li>
|
||
<li>难以用于采样不足的区域</li>
|
||
</ol>
|
||
<p><img src="../../images/eg_point_cloud.png" alt=""></p>
|
||
<h6 id="多边形网格polygon-mesh"><a href="#多边形网格polygon-mesh" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>多边形网格(Polygon Mesh)</h6>
|
||
<ol>
|
||
<li>存储顶点和多边形(通常是三角形或四边形)</li>
|
||
<li>更易于进行处理/模拟，自适应采样</li>
|
||
<li>更复杂的数据结构</li>
|
||
<li>图形中最常见的表示形式</li>
|
||
</ol>
|
||
<p><img src="../../images/eg_polygon_mesh.png" alt=""></p>
|
||
<hr>
|
||
<h3 id="表现形式应根据目标几何模型选择最合适的类型没有最好的表现形式"><a href="#表现形式应根据目标几何模型选择最合适的类型没有最好的表现形式" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>表现形式应根据目标几何模型选择最合适的类型,没有最好的表现形式</h3>
|
||
<p><img src="../../images/David_Baraff.png" alt=""></p>
|
||
<hr>
|
||
<h3 id="贝塞尔曲线bézier-curves"><a href="#贝塞尔曲线bézier-curves" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>贝塞尔曲线(Bézier Curves)</h3>
|
||
<h4 id="定义"><a href="#定义" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>定义</h4>
|
||
<p>贝塞尔曲线完全由其控制点决定其形状,$n$个控制点对应着$n-1$阶的贝塞尔曲线，并且可以通过递归的方式来绘制.</p>
|
||
<p><img src="../../images/Defining_Bezier_Curve_Tangents.png" alt=""></p>
|
||
<h4 id="de-casteljau-algorithm图形"><a href="#de-casteljau-algorithm图形" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>de Casteljau Algorithm(图形)</h4>
|
||
<p>假设存在三个点(quadratic Bezier)</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step1.png" alt=""></p>
|
||
<p>通过线性插值的方式插入一个点</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step2.png" alt=""></p>
|
||
<p>在另一边也通过同样方式插入一个点</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step3.png" alt=""></p>
|
||
<p>递归重复</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step4.png" alt=""></p>
|
||
<p>对于在$[0,1]$区间的每个t点都使用相同算法进行计算</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step5.png" alt=""></p>
|
||
<p>构造一个三次方贝塞尔曲线需要总共四个输入，都递归使用线性插值</p>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step6.png" alt=""></p>
|
||
<p>可视化算法流程</p>
|
||
<p><img src="../../images/Visualizing_de_Casteljau.png" alt=""></p>
|
||
<h4 id="de-casteljau-algorithm数学公式"><a href="#de-casteljau-algorithm数学公式" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>de Casteljau Algorithm(数学公式)</h4>
|
||
<p>de Casteljau 算法给出各点间金字塔型的变量关系</p>
|
||
<p><img src="../../images/de_Casteljau_pyramid_of.png" alt=""></p>
|
||
<h5 id="推导流程"><a href="#推导流程" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>推导流程</h5>
|
||
<p><img src="../../images/de_Casteljau_Algorithm_step5.png" alt=""></p>
|
||
<p>$$b_0^1(t) = (1 - t)b_0 + tb_1$$
|
||
$$b_1^1(t) = (1 - t)b_1 + tb_2$$
|
||
$$b_0^2(t) = (1 - t)b_0^1 + tb_1^1$$
|
||
$$b_0^2(t) = (1 - t)^2b_0 + 2t(1 - t)b_1 + t^2b_2$$</p>
|
||
<p>进而推得$n$阶贝塞尔曲线的公式为：</p>
|
||
<p>$$b^n(t) = b_0^n(t) = \sum_{j=0}^{n} {b_j B_j^n(t)}$$</p>
|
||
<p>$b^n(t)$:贝塞尔曲线阶级数n(n次向量多项式)</p>
|
||
<p>$b_j$:控制点($R^N$的向量)</p>
|
||
<p>$B_j^n$:伯恩斯坦多项式</p>
|
||
<p>伯恩斯坦多项式(Bernstein polynomial):</p>
|
||
<p>$$B_i^n(t) = \begin{pmatrix} n \\ t \end{pmatrix} t^i(1 - t)^{n - i}$$</p>
|
||
<p>例如$n = 3$时</p>
|
||
<p>我们在三维空间里有下列控制点</p>
|
||
<p>$b_0 = (0,2,3),b_1 = (2,3,5),b_2 = (6,7,9),b_3 = (3,4,5)$</p>
|
||
<p>这些点定义了以下列公式形式的贝塞尔曲线</p>
|
||
<p>$$b^n(t) = b_0(1 - t)^3 + b_1 3t(1 - t)^2 + b_2 3t^2(1 - t) + b_3 t^3$$</p>
|
||
<p>贝塞尔基本函数</p>
|
||
<p><img src="../../images/Cubic_Bezier_Basis_Functions.png" alt=""></p>
|
||
<p>插值端点：$b(0) = b_0;b(1) = b_3$</p>
|
||
<p>与末端线段相切：$b'(0) = 3(b_1 - b_0); b'(1) = 3(b_3 - b_2)$</p>
|
||
<p>能够通过控制点的位置来改变曲线</p>
|
||
<h3 id="分段贝塞尔曲线"><a href="#分段贝塞尔曲线" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>分段贝塞尔曲线</h3>
|
||
<p>出现原因：高阶贝塞尔曲线有多个控制点，难以控制曲线形状</p>
|
||
<p>对策: 将多个低阶的贝塞尔曲线相连，构造为分段贝塞尔曲线</p>
|
||
<p>常用于：字体，路径，插画，主题演讲</p>
|
||
<p>样例：</p>
|
||
<p><img src="../../images/Demon_Piecewise_Cubic_Bezier_Curve.png" alt=""></p>
|
||
<h4 id="组合计算"><a href="#组合计算" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>组合计算</h4>
|
||
<p>要将一下两条贝塞尔曲线组合在一起则</p>
|
||
<p>$$a:[k,k+1] \rightarrow IR^N$$
|
||
$$b:[k+1,k+2] \rightarrow IR^N$$</p>
|
||
<p><img src="../../images/Continuity_Piecewise_Bezier_step1.png" alt=""></p>
|
||
<p>$c^0$处的连续性：$a_n = b_0$</p>
|
||
<p><img src="../../images/Continuity_Piecewise_Bezier_step2.png" alt=""></p>
|
||
<p>$c^1$处的连续性：$a_n = b_0 = \frac{1}{2}(a_{n-1} + b_1)$</p>
|
||
<p><img src="../../images/Continuity_Piecewise_Bezier_step3.png" alt=""></p>
|
||
<hr>
|
||
<h3 id="贝塞尔表面bézier-surfaces"><a href="#贝塞尔表面bézier-surfaces" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>贝塞尔表面(Bézier Surfaces)</h3>
|
||
<p>对于一个双立方贝塞尔表面贴片</p>
|
||
<p>输入：$4 \times 4 $ 控制点</p>
|
||
<p>输出：由$[0,1]^2$ 参数化的2D平面</p>
|
||
<p><img src="../../images/bezier_surface_eg.png" alt=""></p>
|
||
<h4 id="计算方法"><a href="#计算方法" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>计算方法</h4>
|
||
<p>目标：计算相对于(u,v)的平面位置</p>
|
||
<ol>
|
||
<li>使用 de Casteljau 算法来计算四条贝塞尔曲线上各自的U,这将会为"移动"贝塞尔曲线提供4个有效的控制点</li>
|
||
<li>使用一阶 de Casteljau 算法来计算"移动"曲线上的点V</li>
|
||
</ol>
|
||
<p><img src="../../images/evaluation_of_bezier_surface.png" alt=""></p>
|
||
<p>可视化计算流程：</p>
|
||
<p><img src="../../images/visual_evaluation_of_bezier_surface.png" alt=""></p>
|
||
<h3 id="曲面处理mesh-operations"><a href="#曲面处理mesh-operations" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>曲面处理(Mesh Operations)</h3>
|
||
<h4 id="曲面细分mesh-subdivision"><a href="#曲面细分mesh-subdivision" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>曲面细分(Mesh Subdivision)</h4>
|
||
<p>目的：提高分辨率</p>
|
||
<p><img src="../../images/eg_mesh_subdivision.png" alt=""></p>
|
||
<p>通常做法：</p>
|
||
<ol>
|
||
<li>创建更多三角形(顶点)</li>
|
||
<li>更新它们的位置</li>
|
||
</ol>
|
||
<h5 id="loop-subdivision"><a href="#loop-subdivision" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>Loop Subdivision</h5>
|
||
<ol>
|
||
<li>将每个三角形划分为4个三角形</li>
|
||
<li>根据权重更新顶点的位置(新旧顶点各自以不同方式进行更新)</li>
|
||
</ol>
|
||
<p>新顶点：</p>
|
||
<p><img src="../../images/loop_subdivision_new.png" alt=""></p>
|
||
<p>白点即为新顶点，其位置为周围四个顶点的权重之和</p>
|
||
<p>旧顶点：</p>
|
||
<p><img src="../../images/loop_subdivision_old.png" alt=""></p>
|
||
<p>白色旧顶点也是自身及邻接顶点的权重之和，权重的设置与旧顶点度数关联</p>
|
||
<h5 id="catmull-clark-subdivision"><a href="#catmull-clark-subdivision" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>Catmull-Clark Subdivision</h5>
|
||
<p><img src="../../images/catmull-clark_subdivision1.png" alt=""></p>
|
||
<p>定义：</p>
|
||
<ol>
|
||
<li>所有非四边形的面都称为Non-quad face</li>
|
||
<li>所有度不为4的点称为奇异点</li>
|
||
<li>每次细分时在每个面中添加一个点，每条边的中点也都添加一个点，面上的新顶点连接所有边上的新顶点</li>
|
||
</ol>
|
||
<p>第一次细分后结果：</p>
|
||
<p><img src="../../images/catmull-clark_subdivision2.png" alt=""></p>
|
||
<p>特点：</p>
|
||
<ol>
|
||
<li>非四边形面的数量与奇异点相同，即现在共有$2+2=4$个</li>
|
||
<li>奇异点的度数与原来所在面的边数相等，即这里为3度</li>
|
||
<li>第一次细分后所有的面都会变成四边形，且后续奇异点数目不再增加</li>
|
||
</ol>
|
||
<p>Catmull-Clark 顶点更新规则</p>
|
||
<p><img src="../../images/Catmull-Clark_Vertex_Update_Rules.png" alt=""></p>
|
||
<h5 id="收敛性整体形状和折痕"><a href="#收敛性整体形状和折痕" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>收敛性：整体形状和折痕</h5>
|
||
<p><img src="../../images/convergence_of_loop_and_catmull.png" alt=""></p>
|
||
<h4 id="曲面简化mesh-simplification"><a href="#曲面简化mesh-simplification" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>曲面简化(Mesh Simplification)</h4>
|
||
<p>目的：降低分辨率的同时尽量保持形状/外观</p>
|
||
<p><img src="../../images/eg_mesh_simplification.png" alt=""></p>
|
||
<h5 id="边坍缩"><a href="#边坍缩" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>边坍缩</h5>
|
||
<p>边坍缩是曲面简化的常用方法，如上图所示将一条边的两个顶点合成为一个顶点，出于尽量保持形状的目的，需要正确选择不影响或影响最小的边进行坍缩，由此引入二次误差度量(Quadric Error Metrics)</p>
|
||
<p><img src="../../images/Collapsing_An_Edge.png" alt=""></p>
|
||
<h6 id="二次误差度量quadric-error-metrics"><a href="#二次误差度量quadric-error-metrics" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>二次误差度量(Quadric Error Metrics)</h6>
|
||
<p><img src="../../images/Quadric_Error_Metrics.png" alt=""></p>
|
||
<p>坍缩之后蓝色新顶点所在位置与原来各个平面的垂直距离之和，如此误差最小则整个模型样貌修改一定程度也会较小</p>
|
||
<h5 id="曲面简化算法流程"><a href="#曲面简化算法流程" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>曲面简化算法流程</h5>
|
||
<ol>
|
||
<li>为模型每条边赋值，其值为坍缩后代替老顶点产生的新顶点所得到的最小二次误差</li>
|
||
<li>选取权重最小的边做坍缩，新顶点的位置为原计算得出使二次误差值最小的位置</li>
|
||
<li>坍缩之后，会改动与之相连的其他边，更新这些边的权值</li>
|
||
<li>重复步骤，直到符合终止条件</li>
|
||
</ol>
|
||
<p>符合贪心算法标准，无法获得最优解，但效果依旧合适</p>
|
||
<p><img src="../../images/eg_Quadric_Error_Mesh_Simplification.png" alt=""></p>
|
||
<hr>
|
||
<h4 id="纹理映射shadow-mapping"><a href="#纹理映射shadow-mapping" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>纹理映射(Shadow Mapping)</h4>
|
||
<p>Shadow Mapping是一种基于图像的算法</p>
|
||
<ol>
|
||
<li>阴影计算期间无需进行几何体计算</li>
|
||
<li>必须进行反走样处理</li>
|
||
<li>不在阴影中的点必须同时在灯光和相机范围内</li>
|
||
</ol>
|
||
<h5 id="计算流程"><a href="#计算流程" class="anchor-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon anchor-icon"><path d="M326.612 185.391c59.747 59.809 58.927 155.698.36 214.59-.11.12-.24.25-.36.37l-67.2 67.2c-59.27 59.27-155.699 59.262-214.96 0-59.27-59.26-59.27-155.7 0-214.96l37.106-37.106c9.84-9.84 26.786-3.3 27.294 10.606.648 17.722 3.826 35.527 9.69 52.721 1.986 5.822.567 12.262-3.783 16.612l-13.087 13.087c-28.026 28.026-28.905 73.66-1.155 101.96 28.024 28.579 74.086 28.749 102.325.51l67.2-67.19c28.191-28.191 28.073-73.757 0-101.83-3.701-3.694-7.429-6.564-10.341-8.569a16.037 16.037 0 0 1-6.947-12.606c-.396-10.567 3.348-21.456 11.698-29.806l21.054-21.055c5.521-5.521 14.182-6.199 20.584-1.731a152.482 152.482 0 0 1 20.522 17.197zM467.547 44.449c-59.261-59.262-155.69-59.27-214.96 0l-67.2 67.2c-.12.12-.25.25-.36.37-58.566 58.892-59.387 154.781.36 214.59a152.454 152.454 0 0 0 20.521 17.196c6.402 4.468 15.064 3.789 20.584-1.731l21.054-21.055c8.35-8.35 12.094-19.239 11.698-29.806a16.037 16.037 0 0 0-6.947-12.606c-2.912-2.005-6.64-4.875-10.341-8.569-28.073-28.073-28.191-73.639 0-101.83l67.2-67.19c28.239-28.239 74.3-28.069 102.325.51 27.75 28.3 26.872 73.934-1.155 101.96l-13.087 13.087c-4.35 4.35-5.769 10.79-3.783 16.612 5.864 17.194 9.042 34.999 9.69 52.721.509 13.906 17.454 20.446 27.294 10.606l37.106-37.106c59.271-59.259 59.271-155.699.001-214.959z"/></svg></a>计算流程</h5>
|
||
<p>阴影映射总体需要两个pass</p>
|
||
<p>Pass1:render from light</p>
|
||
<ol>
|
||
<li>获得从光源视角得到的深度图像</li>
|
||
</ol>
|
||
<p><img src="../../images/render_from_light.png" alt=""></p>
|
||
<p>Pass2:render from eye</p>
|
||
<ol>
|
||
<li>从观看视角(相机视角)获得带有深度的标准图像</li>
|
||
<li>将观看视角中的可见点投影回光源
|
||
<ol>
|
||
<li>光源和观看视角的下的深度相同时为可见</li>
|
||
<li>光源和观看视角下的深度不相同则为被阻挡</li>
|
||
</ol>
|
||
</li>
|
||
</ol>
|
||
<p><img src="../../images/project_to_light.png" alt=""></p>
|
||
<hr>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
</article>
|
||
|
||
|
||
|
||
|
||
<div class="updated-badge-container">
|
||
<span title="Updated @ 2023-06-18 21:19:01 CST" style="cursor:help">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" width="130" height="20" class="updated-badge"><linearGradient id="b" x2="0" y2="100%"><stop offset="0" stop-color="#bbb" stop-opacity=".1"/><stop offset="1" stop-opacity=".1"/></linearGradient><clipPath id="a"><rect width="130" height="20" rx="3" fill="#fff"/></clipPath><g clip-path="url(#a)"><path class="updated-badge-left" d="M0 0h55v20H0z"/><path class="updated-badge-right" d="M55 0h75v20H55z"/><path fill="url(#b)" d="M0 0h130v20H0z"/></g><g fill="#fff" text-anchor="middle" font-size="110"><text x="285" y="150" fill="#010101" fill-opacity=".3" textLength="450" transform="scale(.1)">updated</text><text x="285" y="140" textLength="450" transform="scale(.1)">updated</text><text x="915" y="150" fill="#010101" fill-opacity=".3" textLength="650" transform="scale(.1)">2023-06-18</text><text x="915" y="140" textLength="650" transform="scale(.1)">2023-06-18</text></g></svg>
|
||
</span></div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<ul class="post-nav">
|
||
|
||
<li class="post-nav-prev">
|
||
<a href="/computergraphic/whitted-style%E5%85%89%E7%BA%BF%E8%BF%BD%E8%B8%AA/" rel="prev">< Whitted-style光线追踪(Whitted-style Ray Tracing)</a>
|
||
</li>
|
||
|
||
|
||
<li class="post-nav-next">
|
||
<a href="/computergraphic/%E7%BA%B9%E7%90%86/" rel="next">纹理 ></a>
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</div>
|
||
</main>
|
||
|
||
|
||
|
||
<div id="back-to-top" class="back-to-top">
|
||
<a href="#"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon arrow-up"><path d="M34.9 289.5l-22.2-22.2c-9.4-9.4-9.4-24.6 0-33.9L207 39c9.4-9.4 24.6-9.4 33.9 0l194.3 194.3c9.4 9.4 9.4 24.6 0 33.9L413 289.4c-9.5 9.5-25 9.3-34.3-.4L264 168.6V456c0 13.3-10.7 24-24 24h-32c-13.3 0-24-10.7-24-24V168.6L69.2 289.1c-9.3 9.8-24.8 10-34.3.4z"/></svg></a>
|
||
</div>
|
||
|
||
|
||
|
||
<footer id="footer" class="footer">
|
||
<div class="footer-inner">
|
||
<div class="site-info">© 2022–2023 <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon footer-icon"><path d="M462.3 62.6C407.5 15.9 326 24.3 275.7 76.2L256 96.5l-19.7-20.3C186.1 24.3 104.5 15.9 49.7 62.6c-62.8 53.6-66.1 149.8-9.9 207.9l193.5 199.8c12.5 12.9 32.8 12.9 45.3 0l193.5-199.8c56.3-58.1 53-154.3-9.8-207.9z"/></svg> InkSoul</div><div class="powered-by">Powered by <a href="https://github.com/gohugoio/hugo" target="_blank" rel="noopener">Hugo</a> | Theme is <a href="https://github.com/reuixiy/hugo-theme-meme" target="_blank" rel="noopener">MemE</a></div><div class="site-copyright"><a href="https://creativecommons.org/licenses/by-nc-sa/4.0/deed.zh" target="_blank" rel="noopener">CC BY-NC-SA 4.0</a></div><div class="custom-footer"><a href="https://beian.miit.gov.cn" target="_blank" rel="noopener">闽ICP备2022009452号-1</a></div>
|
||
|
||
|
||
|
||
<ul class="socials"><li class="socials-item">
|
||
<a href="mailto:qingci30@163.com" target="_blank" rel="external noopener" title="Email"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon social-icon"><path d="M464 64H48C21.49 64 0 85.49 0 112v288c0 26.51 21.49 48 48 48h416c26.51 0 48-21.49 48-48V112c0-26.51-21.49-48-48-48zm0 48v40.805c-22.422 18.259-58.168 46.651-134.587 106.49-16.841 13.247-50.201 45.072-73.413 44.701-23.208.375-56.579-31.459-73.413-44.701C106.18 199.465 70.425 171.067 48 152.805V112h416zM48 400V214.398c22.914 18.251 55.409 43.862 104.938 82.646 21.857 17.205 60.134 55.186 103.062 54.955 42.717.231 80.509-37.199 103.053-54.947 49.528-38.783 82.032-64.401 104.947-82.653V400H48z"/></svg></a>
|
||
</li><li class="socials-item">
|
||
<a href="https://github.com/ink-soul" target="_blank" rel="external noopener" title="GitHub"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24" class="icon social-icon"><path d="M12 .297c-6.63 0-12 5.373-12 12 0 5.303 3.438 9.8 8.205 11.385.6.113.82-.258.82-.577 0-.285-.01-1.04-.015-2.04-3.338.724-4.042-1.61-4.042-1.61C4.422 18.07 3.633 17.7 3.633 17.7c-1.087-.744.084-.729.084-.729 1.205.084 1.838 1.236 1.838 1.236 1.07 1.835 2.809 1.305 3.495.998.108-.776.417-1.305.76-1.605-2.665-.3-5.466-1.332-5.466-5.93 0-1.31.465-2.38 1.235-3.22-.135-.303-.54-1.523.105-3.176 0 0 1.005-.322 3.3 1.23.96-.267 1.98-.399 3-.405 1.02.006 2.04.138 3 .405 2.28-1.552 3.285-1.23 3.285-1.23.645 1.653.24 2.873.12 3.176.765.84 1.23 1.91 1.23 3.22 0 4.61-2.805 5.625-5.475 5.92.42.36.81 1.096.81 2.22 0 1.606-.015 2.896-.015 3.286 0 .315.21.69.825.57C20.565 22.092 24 17.592 24 12.297c0-6.627-5.373-12-12-12"/></svg></a>
|
||
</li><li class="socials-item">
|
||
<a href="http://gitea.inksoul.top/inksoul" target="_blank" rel="external noopener" title="gitea"><svg class="icon social-icon" fill="none" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><path d="M4.209 4.603c-.247 0-.525.02-.84.088-.333.07-1.28.283-2.054 1.027C-.403 7.25.035 9.685.089 10.052c.065.446.263 1.687 1.21 2.768 1.749 2.141 5.513 2.092 5.513 2.092s.462 1.103 1.168 2.119c.955 1.263 1.936 2.248 2.89 2.367 2.406 0 7.212-.004 7.212-.004s.458.004 1.08-.394c.535-.324 1.013-.893 1.013-.893s.492-.527 1.18-1.73c.21-.37.385-.729.538-1.068 0 0 2.107-4.471 2.107-8.823-.042-1.318-.367-1.55-.443-1.627-.156-.156-.366-.153-.366-.153s-4.475.252-6.792.306c-.508.011-1.012.023-1.512.027v4.474l-.634-.301c0-1.39-.004-4.17-.004-4.17-1.107.016-3.405-.084-3.405-.084s-5.399-.27-5.987-.324c-.187-.011-.401-.032-.648-.032zm.354 1.832h.111s.271 2.269.6 3.597C5.549 11.147 6.22 13 6.22 13s-.996-.119-1.641-.348c-.99-.324-1.409-.714-1.409-.714s-.73-.511-1.096-1.52C1.444 8.73 2.021 7.7 2.021 7.7s.32-.859 1.47-1.145c.395-.106.863-.12 1.072-.12zm8.33 2.554c.26.003.509.127.509.127l.868.422-.529 1.075a.686.686 0 0 0-.614.359.685.685 0 0 0 .072.756l-.939 1.924a.69.69 0 0 0-.66.527.687.687 0 0 0 .347.763.686.686 0 0 0 .867-.206.688.688 0 0 0-.069-.882l.916-1.874a.667.667 0 0 0 .237-.02.657.657 0 0 0 .271-.137 8.826 8.826 0 0 1 1.016.512.761.761 0 0 1 .286.282c.073.21-.073.569-.073.569-.087.29-.702 1.55-.702 1.55a.692.692 0 0 0-.676.477.681.681 0 1 0 1.157-.252c.073-.141.141-.282.214-.431.19-.397.515-1.16.515-1.16.035-.066.218-.394.103-.814-.095-.435-.48-.638-.48-.638-.467-.301-1.116-.58-1.116-.58s0-.156-.042-.27a.688.688 0 0 0-.148-.241l.516-1.062 2.89 1.401s.48.218.583.619c.073.282-.019.534-.069.657-.24.587-2.1 4.317-2.1 4.317s-.232.554-.748.588a1.065 1.065 0 0 1-.393-.045l-.202-.08-4.31-2.1s-.417-.218-.49-.596c-.083-.31.104-.691.104-.691l2.073-4.272s.183-.37.466-.497a.855.855 0 0 1 .35-.077z" ></path></svg></a>
|
||
</li><li class="socials-item">
|
||
<a href="https://space.bilibili.com/262369930" target="_blank" rel="external noopener" title="bilibili"><svg class="icon social-icon" viewBox="0 0 24 24" xmlns="http://www.w3.org/2000/svg"><path d="M17.813 4.653h.854c1.51.054 2.769.578 3.773 1.574 1.004.995 1.524 2.249 1.56 3.76v7.36c-.036 1.51-.556 2.769-1.56 3.773s-2.262 1.524-3.773 1.56H5.333c-1.51-.036-2.769-.556-3.773-1.56S.036 18.858 0 17.347v-7.36c.036-1.511.556-2.765 1.56-3.76 1.004-.996 2.262-1.52 3.773-1.574h.774l-1.174-1.12a1.234 1.234 0 0 1-.373-.906c0-.356.124-.658.373-.907l.027-.027c.267-.249.573-.373.92-.373.347 0 .653.124.92.373L9.653 4.44c.071.071.134.142.187.213h4.267a.836.836 0 0 1 .16-.213l2.853-2.747c.267-.249.573-.373.92-.373.347 0 .662.151.929.4.267.249.391.551.391.907 0 .355-.124.657-.373.906zM5.333 7.24c-.746.018-1.373.276-1.88.773-.506.498-.769 1.13-.786 1.894v7.52c.017.764.28 1.395.786 1.893.507.498 1.134.756 1.88.773h13.334c.746-.017 1.373-.275 1.88-.773.506-.498.769-1.129.786-1.893v-7.52c-.017-.765-.28-1.396-.786-1.894-.507-.497-1.134-.755-1.88-.773zM8 11.107c.373 0 .684.124.933.373.25.249.383.569.4.96v1.173c-.017.391-.15.711-.4.96-.249.25-.56.374-.933.374s-.684-.125-.933-.374c-.25-.249-.383-.569-.4-.96V12.44c0-.373.129-.689.386-.947.258-.257.574-.386.947-.386zm8 0c.373 0 .684.124.933.373.25.249.383.569.4.96v1.173c-.017.391-.15.711-.4.96-.249.25-.56.374-.933.374s-.684-.125-.933-.374c-.25-.249-.383-.569-.4-.96V12.44c.017-.391.15-.711.4-.96.249-.249.56-.373.933-.373Z"/></svg></a>
|
||
</li></ul>
|
||
|
||
|
||
|
||
|
||
|
||
</div>
|
||
</footer>
|
||
|
||
|
||
</div>
|
||
<script>
|
||
if ('serviceWorker' in navigator) {
|
||
window.addEventListener('load', function() {
|
||
navigator.serviceWorker.register('\/sw.js');
|
||
});
|
||
}
|
||
</script>
|
||
|
||
|
||
|
||
<link rel="stylesheet" href="https://fastly.jsdelivr.net/npm/katex@0.13.0/dist/katex.min.css" integrity="sha256-gPJfuwTULrEAAcI3X4bALVU/2qBU+QY/TpoD3GO+Exw=" crossorigin="anonymous">
|
||
|
||
<script>
|
||
if (typeof renderMathInElement === 'undefined') {
|
||
var getScript = (options) => {
|
||
var script = document.createElement('script');
|
||
script.defer = true;
|
||
script.crossOrigin = 'anonymous';
|
||
Object.keys(options).forEach((key) => {
|
||
script[key] = options[key];
|
||
});
|
||
document.body.appendChild(script);
|
||
};
|
||
getScript({
|
||
src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/katex.min.js',
|
||
integrity: 'sha256-YTW9cMncW/ZQMhY69KaUxIa2cPTxV87Uh627Gf5ODUw=',
|
||
onload: () => {
|
||
getScript({
|
||
src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/contrib/mhchem.min.js',
|
||
integrity: 'sha256-yzSfYeVsWJ1x+2g8CYHsB/Mn7PcSp8122k5BM4T3Vxw=',
|
||
onload: () => {
|
||
getScript({
|
||
src: 'https:\/\/fastly.jsdelivr.net/npm/katex@0.13.0/dist/contrib/auto-render.min.js',
|
||
integrity: 'sha256-fxJzNV6hpc8tgW8tF0zVobKa71eTCRGTgxFXt1ZpJNM=',
|
||
onload: () => {
|
||
renderKaTex();
|
||
}
|
||
});
|
||
}
|
||
});
|
||
}
|
||
});
|
||
} else {
|
||
renderKaTex();
|
||
}
|
||
function renderKaTex() {
|
||
renderMathInElement(
|
||
document.body,
|
||
{
|
||
delimiters: [
|
||
{left: "$$", right: "$$", display: true},
|
||
{left: "\\[", right: "\\]", display: true},
|
||
{left: "$", right: "$", display: false},
|
||
{left: "\\(", right: "\\)", display: false}
|
||
]
|
||
}
|
||
);
|
||
}
|
||
</script>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<script src="https://fastly.jsdelivr.net/npm/medium-zoom@latest/dist/medium-zoom.min.js"></script>
|
||
|
||
<script>
|
||
let imgNodes = document.querySelectorAll('div.post-body img');
|
||
imgNodes = Array.from(imgNodes).filter(node => node.parentNode.tagName !== "A");
|
||
|
||
mediumZoom(imgNodes, {
|
||
background: 'hsla(var(--color-bg-h), var(--color-bg-s), var(--color-bg-l), 0.95)'
|
||
})
|
||
</script>
|
||
|
||
|
||
|
||
|
||
<script src="https://fastly.jsdelivr.net/npm/instant.page@5.1.0/instantpage.min.js" type="module" defer></script>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</body>
|
||
</html>
|