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<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" />
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<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
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<meta property="og:locale" content="ko-KR" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" />
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<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
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<li><a href="./ja-JP/index.html">日本語</a> <span class="localisation-progress">(24%)</span></li>
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<meta property="og:locale" content="en-GB" />
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<meta property="og:published_time" content="Fri Sep 18 2020 00:00:00 +00:00" />
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<meta property="og:tag" content="Bézier Curves" />

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<meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" />
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<meta property="og:description" content="" />
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<meta property="og:updated_time" content="" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />

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<atom:link href="https://pomax.github.io/bezierinfo" rel="self"></atom:link>
<description>News updates for the <a href="https://pomax.github.io/bezierinfo">primer on Bézier Curves</a> by Pomax</description>
<language>en-GB</language>
<lastBuildDate>Sat Sep 03 2022 16:32:04 +00:00</lastBuildDate>
<lastBuildDate>Sat Sep 03 2022 16:36:08 +00:00</lastBuildDate>
<image>
<url>https://pomax.github.io/bezierinfo/images/og-image.png</url>
<title>A Primer on Bézier Curves</title>

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<meta property="og:locale" content="zh-CN" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" />
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<li><a href="zh-CN/index.html#derivatives">Derivatives</a></li>
<li><a href="zh-CN/index.html#pointvectors">Tangents and normals</a></li>
<li><a href="zh-CN/index.html#pointvectors3d">Working with 3D normals</a></li>
<li><a href="zh-CN/index.html#components">Component functions</a></li>
<li><a href="zh-CN/index.html#components">分量函数</a></li>
<li><a href="zh-CN/index.html#extremities">Finding extremities: root finding</a></li>
<li><a href="zh-CN/index.html#boundingbox">Bounding boxes</a></li>
<li><a href="zh-CN/index.html#aligning">Aligning curves</a></li>
@@ -3501,31 +3501,18 @@ generateRMFrames(steps) -> frames:
<div class="nav">
<a href="zh-CN/index.html#pointvectors3d">上一节</a><a href="#toc">目录</a><a href="zh-CN/index.html#extremities">下一节</a>
</div>
<a href="zh-CN/index.html#components">Component functions</a>
<a href="zh-CN/index.html#components">分量函数</a>
</h1>
<p>
One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but
how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on
exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its
"extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever
took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that
function, but this poses a problem, since our function is parametric: every axis has its own function.
当人们开始在自己的程序中使用贝塞尔曲线时，首先遇到的问题之一是：“我虽然知道怎么画曲线，但是怎么确定包围盒？”其实做法颇为直接，但需要知道如何利用一些数学知识得到所需的值。对于包围盒而言，所需的其实并不是曲线本身，而只是曲线的“极值”——曲线的x轴和y轴分量的最小值和最大值。如果还记得微积分的话（前提是学过微积分，否则更难记），那么函数的极值可以用函数的一阶导数所确定，但由于“曲线函数”有不只一个分量，这就产生了一个问题——每个分量都有自己的函数。
</p>
<p>解决办法：对每个分量分别计算导数，再按照原来的分量顺序重新拼在一起。</p>
<p>
The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.
</p>
<p>
Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the
leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost
figures are the component functions for computing the x-axis value, given a value for <i>t</i> (between 0 and 1 inclusive), and the y-axis
value, respectively.
</p>
<p>
If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a
change in the right graph.
以下演示参数化的贝塞尔曲线如何“分解”为两个正常的函数，一个对应于x轴，一个对应于y轴。注意左侧的图像依然是可交互的曲线，但没有标出坐标轴（坐标显示在图中）；中间和右侧的图像是分量函数，分别对应于指定<em>t</em>值（介于0和1之间，含端点）后求出的x轴和y轴分量。
</p>
<p>如果水平移动曲线上的点，那么应当只有中间的图像在变化；同样，如果竖直移动曲线上的点，那么应当只有右侧的图像在变化。</p>
<graphics-element
title="Quadratic Bézier curve components"
title="二次贝塞尔曲线的分量"
width="825"
height="275"
src="./chapters/components/components.js"
@@ -3538,9 +3525,8 @@ generateRMFrames(steps) -> frames:
<img width="825px" height="275px" src="./images/chapters/components/1e6e38f6403dbe4c8b80295a94fc6748.png" loading="lazy" />
<label></label> </fallback-image
></graphics-element>
<p>&nbsp;</p>
<graphics-element
title="Cubic Bézier curve components"
title="三次贝塞尔曲线的分量"
width="825"
height="275"
src="./chapters/components/components.js"