バッとした導入
++ 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。 +
+
+ JSがなくて、画像を表示しています。
+ 
+ JSがなくて、画像を表示しています。
+ + 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！ +
+So what makes a Bézier Curve?
++ Playing with the points for curves may have given you a feel for how + Bézier curves behave, but what are 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: +
++ Bézier curves are the result of + linear interpolations. 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". +
++ 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: +
+
+ + 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: +
+
+ JSがなくて、画像を表示しています。
+ And that brings us to the complicated maths: calculus.
++ 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). +
++ 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. +
+