Today I’ve been mostly amazed by Bézier curves. Cubic curves can be a bit complicated, what with all those unintuitive coefficients and having to choose whether to specify the x value or the y value, but not both. Not so with Bézier curves, though. Here are some amazing facts about them:

• A Bézier curve is defined by four points: the start point, the end point, and two “control points”. But the control points aren’t arbitrary obscure points out in space. They represent tangents that each end of the curve will run along. See the diagram at the top of this page.
• Splitting a Bézier curve into two other Bézier curves is really easy. You just draw lines between your start/end/control points, find the midpoint of each, and you’ve got your start/end/control points of your new curves.
• The maths turns out to be wonderfully neat. You can convert from an equation for accurate drawing to the corresponding polynomial equation without involving nasty cube roots.
• Despite all this elegance, the mathematics was only developed in 1962. And there was me thinking all the world’s tidy maths was done in Ye Olden Days with bewigged mathematicians getting ink all over their frilly cuffs.

Finally, here’s a little demo that lets you move the points around.

Tags: mathematics, beziercurves