The Bézier Curve

When you look at Michael Craig-Martin's work that has been drawn on a computer, you might be tempted to emulate the clean graphic style of the images. If so you would most likely go into a software program such as Illustrator in order to do the drawing, where you will have to use what are in computer terms called 'paths' to draw your lines. These lines are aesthetically very pleasing to the eye and of course there is a reason for this. They are made up of what are now called Bézier Curves and the study of the underlying mathematics of these curves was first undertaken in 1959 by mathematician Paul de Casteljau who was working for Citroen. Car designers are always looking for smooth curves that seem to unfold naturally and de Casteljau wanted to see if there was a stable method to evaluate certain types of curves that had a close association with straight lines. I.e. curves that would not visually 'jump' or appear to perceptually move too quickly from a straight into a curve. You will have all had to look at tangents when doing geometry at school, straight lines that touch a curve at a point, but if extended do not cross it at that point. The important issue is that as the tangent line passes through the point where the line and the curve meet, the tangent line is "going in the same direction" as the curve, and is therefore the best straight-line approximation to the curve at that point. I.e. at that point the straight line is as close as possible to the curve. 

Another French mathematician, working this time for Renault would however give his name to this type of curve. In the late 1960s Pierre Bézier was looking at how computers could be used to aid the drawing of smooth curves in car design. He decided to use de Casteljau's equations as a method of computer drawing because curves on computers have to be defined by mathematical equations. The situation of course now is that most vector graphics software packages include a pen tool for drawing paths with Bézier curves. (A vector has direction as well as magnitude, so you can determine the position of one point in space relative to another)

What actually happens when using Bézier curves in vector graphics is that they are split up into segments to make sure that the curve is flat enough to be drawn. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. I.e. visual jumps in a curved line's appearance. See how curves can be developed from tangents below.  

For those of you into the maths; given points P₀ and P₁, a linear Bézier curve is simply a straight line between those two points. The curve is given by

So this is why those curves in Illustrator look so 'right', they have both a strong mathematical basis and were chosen by car designers. 

The drawing above is not by Michael Craig-Martin, it is a drawing done by a designer working for Renault, but when you compare with the drawing of the paint roller below that is by Michael Craig-Martin you can see that they belong to the same visual family. 

Learn how to use the pen tool in PhotoShop here. Then you can practice using Bézier curves all day.

See also:

Euler spirals and road design

Other mathematical forms