Bézier Curves

by G. Adam Stanislav

If you search the Internet for Bézier curves, you will obtain a list of many, many pages that discuss the topic of Bézier curves. But chances are they will come in two varieties: A large number of pages that talk about Bézier curves without really explaining their nature, and a small number of pages that offer an in-depth explanation of Bézier curves but that only professional mathematicians can understand.

Because of that, for the first time in history many designers and graphic artists are using tools extensively but often without understanding those tools. These tools seem like magic. It seems like we can do just about anything with them. Take, for example, Postscript fonts. They use Bézier curves to produce just about any type of typeface previous generations of designers developed the hard way.

Before computer graphics made it possible to accomplish just about anything, typeface designers always worked within the limitations as well as opportunities offered by their tools. The letters we find in ancient Rome, for example, were strongly influenced by the fact that they were chiseled in stone. Chinese calligraphy owes its looks to its use of the brush. And the lettering of the books written by the medieval monks reflects their use of a quill.

The truth is that Bézier curves have their limitations and their opportunities as well. If we understand Bézier curves more than just moving a control point to see how it changes the shape of the curve, we can unleash our creativity in ways we could not before.

On these pages I hope to present more than a cursory explanation of Bézier curves but, hopefully, in a way that anyone with just a High School level of mathematics can understand. I am going to assume that you know what y = ax³ + bx² + cx + d means, but not necessarily can tell the difference between that and P(u) = Ku³ + Lu² + Mu + u. If you do know the difference, I hope my explanation shall not bore you too much.

A Bézier curve A circle

Throughout these pages, you will find illustrations like those above. Though you may see them as bitmaps, they are Postscript images. Indeed, they are converted to a bitmap from their Postscript code on the fly by peps every time you load them. If you click on any of them, you can download their original Postscript code.

I have posted the first of these pages back in 1998, and then have done nothing till October of 2005. I have decided to start adding everything described above in November of 2005. So, if you read this near November 2005, it is still a work in progress. But it is something I am actively working on, so new pages will appear here. Here is what we have or are working on so far (if you can click it, it is there now, if not, it is coming soon):

Copyright © 2005 G. Adam Stanislav
All rights reserved


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