A Primer on Bézier Curves

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If you have JavaScript disabled, you'll have to enable it, as this book heavily relies on JS rendering, both for the base content (it's been written as a React application) and all the interactive graphics, which rely on JS not just for the user interaction but also for the live-drawing (none of the graphics in this book are flat images, they're all live-rendered).

Preface

In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The first of these are as easy to draw as they are easy to make a computer draw. Give a computer the first and last point in the line, and BAM! straight line. No questions asked.

Curves, however, are a much bigger problem. While we can draw curves with ridiculous ease freehand, computers are a bit handicapped in that they can't draw curves unless there is a mathematical function that describes how it should be drawn. In fact, they even need this for straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned, all lines are "functions", regardless of whether they're straight or curves. However, that does mean that we need to come up with fast-to-compute functions that lead to nice looking curves on a computer. There's a number of these, and in this article we'll focus on a particular function that has received quite a bit of attention, and is used in pretty much anything that can draw curves: "Bézier" curves

They're named after Pierre Bézier, who is principally responsible for getting them known to the world as a curve well-suited for design work (working for Renault and publishing his investigations in 1962), although he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician Paul de Casteljau was first, investigating the nature of these curves in 1959 while working at Citroën, coming up with a really elegant way of figuring out how to draw them. However, de Casteljau did not publish his work, making the question "who was first" hard to answer in any absolute sense. Or is it? Bézier curves are, at their core, "Bernstein polynomials", a family of mathematical functions investigated by Sergei Natanovich Bernstein, with publications on them at least as far back as 1912. Anyway, that's mostly trivia, what you are more likely to care about is that these curves are handy: you can link up multiple Bézier curves so that the combination looks like a single curve. If you've ever drawn Photoshop "paths" or worked with vector drawing programs like Flash, Illustrator or Inkscape, those curves you've been drawing are Bézier curves.

So, what if you need to program them yourself? What are the pitfalls? How do you draw them? What are the bounding boxes, how do you determine intersections, how can you extrude a curve, in short: how do you do everything that you might want when you do with these curves? That's what this page is for. Prepare to be mathed!

—Pomax (or in the tweetworld, @TheRealPomax)