I remember the day like it was yesterday. Mrs. Jerkins had called me up to the chalkboard to solve a math problem. As was her practice, she stood scowling off to the side of the chalkboard as I approached. In one hand, she held a rubber-tipped wooden dowel, and in the other, she gripped a chain that hung from an intercom speaker with which she could call the office in one quick, furious pull.
For what seemed like an eternity, I stood with my face a few inches from the chalkboard, staring at the math problem with no clue of how to solve it.
Nervously, I turned to Mrs. Jerkins and asked, “How do I do it?”
In response, she furrowed her brow and angrily slapped the wooden dowel against the chalkboard, saying, “Solve the problem or I call the office!”
I knew I couldn’t give her the correct answer. Frustrated, I began to cry. From that point forward, I loathed math. It scared me.
I won’t pretend to be an expert in math history, but I’ve done enough sleuthing to trace the family history of the modern Bézier curve, which forms the basis of all vector drawing programs in use today. An understanding of this history won’t improve your drawing skills, but it will give you a better appreciation of the tools we use.
So what does the mathematical equation of a Bézier curve look like?
I asked Bill Casselman, professor of mathematics at the University of British Columbia, to give us a peek at a basic Bézier curve and the math behind the art
I think it’s safe to say you’d have an easier time learning to speak Klingon than trying to wrap your brain around the math required to create a Bézier curve. And thanks to Pierre Bézier, you’ll never have to. All you really need to know is that vector art is made up of anchor points and paths and that a Bézier curve is any segment of that path between two
anchor points that requires a curved shape. One piece of art can have thousands of Bézier curves in it, as shown in Figure 1.5.
Odds are good that if your vector drawing has curves in it, you’ll be using Bézier curves to build it. A Bézier curve will have handlebars that protrude out of the various anchor points in your design. You use these to control and manipulate the curves so you can create the exact shapes you need for your design. The more organic and free-form your design is, the more likely you’ll need to manipulate Bézier curves to build the vector shapes. It’s impossible to get elegant and graceful curves without them (Figure 1.6).
That said, you won’t need to use Bézier curves for every project.
For example, if you’re creating an image that’s chunky and graphic—without smooth curves—you can use just anchor points and paths.
I didn’t need to grab the handle bars at all when creating Figure 1.7 (that said, see the Field Notes at the end of this chapter).
Knowing when to use a Bézier curve and when not to has a lot to do with what you’re creating. In Chapter 6, “Rules of Creative Engagement,” we’ll go into more detail about vector build methods and discuss how they can help or hinder your final art.
The use of Bézier curves in vector-based graphic programs has transformed our industry. We can now take our pen and paper ideas and build them precisely using digital tools.
This method allows us to scale our work to any size without degrading its quality and makes repurposing our work easier than ever before.
It was math that created the Bézier curve, but it was artists (many of whom were likely math-phobic) who took those curves and who can now use them to tell fantastic visual stories.
It’s a beautiful irony, and for that I say, “Viva Bézier!”