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<!DOCTYPE html>
<html lang="en-GB">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>A Primer on Bézier Curves</title>
<link rel="shortcut icon" href="favicon.png" type="image/png" />
<base href=".." />
<!-- opengraph information -->
<meta property="og:title" content="A Primer on Bézier Curves" />
<meta
property="og:image"
content="https://pomax.github.io/bezierinfo/images/og-image.png"
/>
<meta property="og:type" content="text" />
<meta property="og:url" content="https://pomax.github.io/bezierinfo" />
<meta
property="og:description"
content="A detailed explanation of Bézier curves, and how to do the many things that we commonly want to do with them."
/>
<meta property="og:locale" content="en_GB" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13 12:00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
<!-- my own referral/page hit tracker -->
<script src="./lib/site/referrer.js" type="module" async></script>
<!-- the part that makes interactive graphics work: an HTML5 <graphics-element> custom element -->
<script
src="./lib/custom-element/graphics-element.js"
type="module"
async
defer
></script>
<link rel="stylesheet" href="./lib/custom-element/graphics-element.css" />
<!-- page styling -->
<link rel="stylesheet" href="style.css" />
</head>
<body>
<header>
<h1>A Primer on Bézier Curves</h1>
<h2>
A free, online book for when you really need to know how to do Bézier
things.
</h2>
<span>Read this in your own language:</span>
<ul>
<li><a href="../en-GB/index.html">English</a></li>
<li><a href="../ja-JP/index.html">日本語</a></li>
<li><a href="../zh-CN/index.html">中文</a></li>
</ul>
</header>
<main>
<section id="preface"></section>
<section id="toc">
<ol>
<li><a href="#introduction">A lightning introduction</a></li>
<li><a href="#whatis">So what makes a Bézier Curve?</a></li>
</ol>
</section>
<section id="changelog"></section>
<section id="chapters">
<section id="introduction">
<h1>A lightning introduction</h1>
<p>
Let's start with the good stuff: when we're talking about Bézier
curves, we're talking about the things that you can see in the
following graphics. They run from some start point to some end
point, with their curvature influenced by one or more "intermediate"
control points. Now, because all the graphics on this page are
interactive, go manipulate those curves a bit: click-drag the
points, and see how their shape changes based on what you do.
</p>
<div class="figure">
<graphics-element
title="Quadratic Bézier curves"
width="200"
height="200"
src="./chapters/introduction/quadratic.js"
>
<fallback-image>
<img
src="./chapters/introduction/quadratic.png"
width="200"
height="200"
/>
<span>Scripts are disabled. Showing fallback image.</span>
</fallback-image>
</graphics-element>
<graphics-element
title="Cubic Bézier curves"
width="200"
height="200"
src="./chapters/introduction/cubic.js"
>
<fallback-image>
<img
src="./chapters/introduction/cubic.png"
width="200"
height="200"
/>
<span>Scripts are disabled. Showing fallback image.</span>
</fallback-image>
</graphics-element>
</div>
<p>
These curves are used a lot in computer aided design and computer
aided manufacturing (CAD/CAM) applications, as well as in graphic
design programs like Adobe Illustrator and Photoshop, Inkscape,
GIMP, etc. and in graphic technologies like scalable vector graphics
(SVG) and OpenType fonts (TTF/OTF). A lot of things use Bézier
curves, so if you want to learn more about them... prepare to get
your learn on!
</p>
</section>
<section id="whatis">
<h1>So what makes a Bézier Curve?</h1>
<p>
Playing with the points for curves may have given you a feel for how
Bézier curves behave, but what <em>are</em> Bézier curves, really?
There are two ways to explain what a Bézier curve is, and they turn
out to be the entirely equivalent, but one of them uses complicated
maths, and the other uses really simple maths. So... let's start
with the simple explanation:
</p>
<p>
Bézier curves are the result of
<a href="https://en.wikipedia.org/wiki/Linear_interpolation"
>linear interpolations</a
>. That sounds complicated but you've been doing linear
interpolation since you were very young: any time you had to point
at something between two other things, you've been applying linear
interpolation. It's simply "picking a point between two points".
</p>
<p>
If we know the distance between those two points, and we want a new
point that is, say, 20% the distance away from the first point (and
thus 80% the distance away from the second point) then we can
compute that really easily:
</p>
<img
className="LaTeX SVG"
src="images/latex/0ee04e4056391f945c1c21c1174a03b5.svg"
width="507px"
height="103px"
/>
<p>
So let's look at that in action: the following graphic is
interactive in that you can use your up and down arrow keys to
increase or decrease the interpolation ratio, to see what happens.
We start with three points, which gives us two lines. Linear
interpolation over those lines gives us two points, between which we
can again perform linear interpolation, yielding a single point. And
that point —and all points we can form in this way for all ratios
taken together— form our Bézier curve:
</p>
<graphics-element
title="Linear Interpolation leading to Bézier curves"
width="200"
height="200"
src="./chapters/whatis/interpolation.js"
>
<fallback-image>
<img src="./chapters/whatis/interpolation.png" />
<span>Scripts are disabled. Showing fallback image.</span>
</fallback-image>
</graphics-element>
<p>And that brings us to the complicated maths: calculus.</p>
<p>
While it doesn't look like that's what we've just done, we actually
just drew a quadratic curve, in steps, rather than in a single go.
One of the fascinating parts about Bézier curves is that they can
both be described in terms of polynomial functions, as well as in
terms of very simple interpolations of interpolations of [...].
That, in turn, means we can look at what these curves can do based
on both "real maths" (by examining the functions, their derivatives,
and all that stuff), as well as by looking at the "mechanical"
composition (which tells us, for instance, that a curve will never
extend beyond the points we used to construct it).
</p>
<p>
So let's start looking at Bézier curves a bit more in depth: their
mathematical expressions, the properties we can derive from them,
and the various things we can do to, and with, Bézier curves.
</p>
</section>
</section>
</main>
<footer>
<!-- ...code goes here... -->
</footer>
</body>
</html>

201
ja-JP/index.html
View File

@@ -1,201 +0,0 @@
<!DOCTYPE html>
<html lang="ja-JP">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>A Primer on Bézier Curves</title>
<link rel="shortcut icon" href="favicon.png" type="image/png" />
<base href=".." />
<!-- opengraph information -->
<meta property="og:title" content="A Primer on Bézier Curves" />
<meta
property="og:image"
content="https://pomax.github.io/bezierinfo/images/og-image.png"
/>
<meta property="og:type" content="text" />
<meta property="og:url" content="https://pomax.github.io/bezierinfo" />
<meta
property="og:description"
content="A detailed explanation of Bézier curves, and how to do the many things that we commonly want to do with them."
/>
<meta property="og:locale" content="en_GB" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13 12:00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
<!-- my own referral/page hit tracker -->
<script src="./lib/site/referrer.js" type="module" async></script>
<!-- the part that makes interactive graphics work: an HTML5 <graphics-element> custom element -->
<script
src="./lib/custom-element/graphics-element.js"
type="module"
async
defer
></script>
<link rel="stylesheet" href="./lib/custom-element/graphics-element.css" />
<!-- page styling -->
<link rel="stylesheet" href="style.css" />
</head>
<body>
<header>
<h1>A Primer on Bézier Curves</h1>
<h2>
A free, online book for when you really need to know how to do Bézier
things.
</h2>
<span>Read this in your own language:</span>
<ul>
<li><a href="../index.html">English</a></li>
<li><a href="../ja-JP/index.html">日本語</a></li>
<li><a href="../zh-CN/index.html">中文</a></li>
</ul>
</header>
<main>
<section id="preface"></section>
<section id="toc">
<ol>
<li><a href="#introduction">バッとした導入</a></li>
<li><a href="#whatis">So what makes a Bézier Curve?</a></li>
</ol>
</section>
<section id="changelog"></section>
<section id="chapters">
<section id="introduction">
<h1>バッとした導入</h1>
<p>
まずは良い例から始めましょう。ベジエ曲線というのは、下の図に表示されているもののことです。ベジエ曲線はある始点からある終点へと延びており、その曲率は1個以上の「中間」制御点に左右されています。さて、このページの図はどれもインタラクティブになっていますので、ここで曲線をちょっと操作してみましょう。点をドラッグしたとき、曲線の形がそれに応じてどう変化するのか、確かめてみてください。
</p>
<div class="figure">
<graphics-element
title="2次のベジエ曲線"
width="200"
height="200"
src="./chapters/introduction/quadratic.js"
>
<fallback-image>
<img
src="./chapters/introduction/quadratic.png"
width="200"
height="200"
/>
<span>JSがなくて、画像を表示しています。</span>
</fallback-image>
</graphics-element>
<graphics-element
title="3次のベジエ曲線"
width="200"
height="200"
src="./chapters/introduction/cubic.js"
>
<fallback-image>
<img
src="./chapters/introduction/cubic.png"
width="200"
height="200"
/>
<span>JSがなくて、画像を表示しています。</span>
</fallback-image>
</graphics-element>
</div>
<p>
ベジエ曲線は、CAD（computer aided designやCAM（computer aided
manufacturing）のアプリケーションで多用されています。もちろん、Adobe
Illustrator・Photoshop・Inkscape・Gimp
などのグラフィックデザインアプリケーションや、SVG（scalable vector
graphics）・OpenTypeフォント（otf/ttf）のようなグラフィック技術でも利用されています。ベジエ曲線はたくさんのものに使われていますので、これについてもっと詳しく学びたいのであれば……さあ、準備しましょう！
</p>
</section>
<section id="whatis">
<h1>So what makes a Bézier Curve?</h1>
<p>
Playing with the points for curves may have given you a feel for how
Bézier curves behave, but what <em>are</em> Bézier curves, really?
There are two ways to explain what a Bézier curve is, and they turn
out to be the entirely equivalent, but one of them uses complicated
maths, and the other uses really simple maths. So... let's start
with the simple explanation:
</p>
<p>
Bézier curves are the result of
<a href="https://en.wikipedia.org/wiki/Linear_interpolation"
>linear interpolations</a
>. That sounds complicated but you've been doing linear
interpolation since you were very young: any time you had to point
at something between two other things, you've been applying linear
interpolation. It's simply "picking a point between two points".
</p>
<p>
If we know the distance between those two points, and we want a new
point that is, say, 20% the distance away from the first point (and
thus 80% the distance away from the second point) then we can
compute that really easily:
</p>
<img
className="LaTeX SVG"
src="images/latex/0ee04e4056391f945c1c21c1174a03b5.svg"
width="507px"
height="103px"
/>
<p>
So let's look at that in action: the following graphic is
interactive in that you can use your up and down arrow keys to
increase or decrease the interpolation ratio, to see what happens.
We start with three points, which gives us two lines. Linear
interpolation over those lines gives us two points, between which we
can again perform linear interpolation, yielding a single point. And
that point —and all points we can form in this way for all ratios
taken together— form our Bézier curve:
</p>
<graphics-element
title="Linear Interpolation leading to Bézier curves"
width="200"
height="200"
src="./chapters/whatis/interpolation.js"
>
<fallback-image>
<img src="./chapters/whatis/interpolation.png" />
<span>JSがなくて、画像を表示しています。</span>
</fallback-image>
</graphics-element>
<p>And that brings us to the complicated maths: calculus.</p>
<p>
While it doesn't look like that's what we've just done, we actually
just drew a quadratic curve, in steps, rather than in a single go.
One of the fascinating parts about Bézier curves is that they can
both be described in terms of polynomial functions, as well as in
terms of very simple interpolations of interpolations of [...].
That, in turn, means we can look at what these curves can do based
on both "real maths" (by examining the functions, their derivatives,
and all that stuff), as well as by looking at the "mechanical"
composition (which tells us, for instance, that a curve will never
extend beyond the points we used to construct it).
</p>
<p>
So let's start looking at Bézier curves a bit more in depth: their
mathematical expressions, the properties we can derive from them,
and the various things we can do to, and with, Bézier curves.
</p>
</section>
</section>
</main>
<footer>
<!-- ...code goes here... -->
</footer>
</body>
</html>

199
zh-CN/index.html
View File

@@ -1,199 +0,0 @@
<!DOCTYPE html>
<html lang="zh-CN">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>A Primer on Bézier Curves</title>
<link rel="shortcut icon" href="favicon.png" type="image/png" />
<base href=".." />
<!-- opengraph information -->
<meta property="og:title" content="A Primer on Bézier Curves" />
<meta
property="og:image"
content="https://pomax.github.io/bezierinfo/images/og-image.png"
/>
<meta property="og:type" content="text" />
<meta property="og:url" content="https://pomax.github.io/bezierinfo" />
<meta
property="og:description"
content="A detailed explanation of Bézier curves, and how to do the many things that we commonly want to do with them."
/>
<meta property="og:locale" content="en_GB" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13 12:00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
<!-- my own referral/page hit tracker -->
<script src="./lib/site/referrer.js" type="module" async></script>
<!-- the part that makes interactive graphics work: an HTML5 <graphics-element> custom element -->
<script
src="./lib/custom-element/graphics-element.js"
type="module"
async
defer
></script>
<link rel="stylesheet" href="./lib/custom-element/graphics-element.css" />
<!-- page styling -->
<link rel="stylesheet" href="style.css" />
</head>
<body>
<header>
<h1>A Primer on Bézier Curves</h1>
<h2>
A free, online book for when you really need to know how to do Bézier
things.
</h2>
<span>Read this in your own language:</span>
<ul>
<li><a href="../index.html">English</a></li>
<li><a href="../ja-JP/index.html">日本語</a></li>
<li><a href="../zh-CN/index.html">中文</a></li>
</ul>
</header>
<main>
<section id="preface"></section>
<section id="toc">
<ol>
<li><a href="#introduction">简单介绍</a></li>
<li><a href="#whatis">So what makes a Bézier Curve?</a></li>
</ol>
</section>
<section id="changelog"></section>
<section id="chapters">
<section id="introduction">
<h1>简单介绍</h1>
<p>
让我们有个好的开始：当我们在谈论贝塞尔曲线的时候，所指的就是你在如下图像看到的东西。它们从某些起点开始，到终点结束，并且受到一个或多个的“中间”控制点的影响。本页面上的图形都是可交互的，你可以拖动这些点，看看这些形状在你的操作下会怎么变化。
</p>
<div class="figure">
<graphics-element
title="二次贝塞尔曲线"
width="200"
height="200"
src="./chapters/introduction/quadratic.js"
>
<fallback-image>
<img
src="./chapters/introduction/quadratic.png"
width="200"
height="200"
/>
<span>脚本已禁用，并显示后备图像。</span>
</fallback-image>
</graphics-element>
<graphics-element
title="三次贝塞尔曲线"
width="200"
height="200"
src="./chapters/introduction/cubic.js"
>
<fallback-image>
<img
src="./chapters/introduction/cubic.png"
width="200"
height="200"
/>
<span>脚本已禁用，并显示后备图像。</span>
</fallback-image>
</graphics-element>
</div>
<p>
这些曲线在计算机辅助设计和计算机辅助制造应用（CAD/CAM）中用的很多。在图形设计软件中也常用到，像Adobe
Illustrator, Photoshop, Inkscape,
Gimp等等。还可以应用在一些图形技术中，像矢量图形（SVG）和OpenType字体（ttf/otf）。许多东西都用到贝塞尔曲线，如果你想更了解它们...准备好继续往下学吧！
</p>
</section>
<section id="whatis">
<h1>So what makes a Bézier Curve?</h1>
<p>
Playing with the points for curves may have given you a feel for how
Bézier curves behave, but what <em>are</em> Bézier curves, really?
There are two ways to explain what a Bézier curve is, and they turn
out to be the entirely equivalent, but one of them uses complicated
maths, and the other uses really simple maths. So... let's start
with the simple explanation:
</p>
<p>
Bézier curves are the result of
<a href="https://en.wikipedia.org/wiki/Linear_interpolation"
>linear interpolations</a
>. That sounds complicated but you've been doing linear
interpolation since you were very young: any time you had to point
at something between two other things, you've been applying linear
interpolation. It's simply "picking a point between two points".
</p>
<p>
If we know the distance between those two points, and we want a new
point that is, say, 20% the distance away from the first point (and
thus 80% the distance away from the second point) then we can
compute that really easily:
</p>
<img
className="LaTeX SVG"
src="images/latex/0ee04e4056391f945c1c21c1174a03b5.svg"
width="507px"
height="103px"
/>
<p>
So let's look at that in action: the following graphic is
interactive in that you can use your up and down arrow keys to
increase or decrease the interpolation ratio, to see what happens.
We start with three points, which gives us two lines. Linear
interpolation over those lines gives us two points, between which we
can again perform linear interpolation, yielding a single point. And
that point —and all points we can form in this way for all ratios
taken together— form our Bézier curve:
</p>
<graphics-element
title="Linear Interpolation leading to Bézier curves"
width="200"
height="200"
src="./chapters/whatis/interpolation.js"
>
<fallback-image>
<img src="./chapters/whatis/interpolation.png" />
<span>脚本已禁用，并显示后备图像。</span>
</fallback-image>
</graphics-element>
<p>And that brings us to the complicated maths: calculus.</p>
<p>
While it doesn't look like that's what we've just done, we actually
just drew a quadratic curve, in steps, rather than in a single go.
One of the fascinating parts about Bézier curves is that they can
both be described in terms of polynomial functions, as well as in
terms of very simple interpolations of interpolations of [...].
That, in turn, means we can look at what these curves can do based
on both "real maths" (by examining the functions, their derivatives,
and all that stuff), as well as by looking at the "mechanical"
composition (which tells us, for instance, that a curve will never
extend beyond the points we used to construct it).
</p>
<p>
So let's start looking at Bézier curves a bit more in depth: their
mathematical expressions, the properties we can derive from them,
and the various things we can do to, and with, Bézier curves.
</p>
</section>
</section>
</main>
<footer>
<!-- ...code goes here... -->
</footer>
</body>
</html>