Pierre Bezier at a confrence on Computer Aided Geometric Design, an obscure branch of mathematics that created most of the algorithms used in every drawing program from AutoCAD to 3D Studio Max, gave a lecture on how he came up with the Bezier Curve, the thing that made him famous among computer graphic junkies.
The story of the Bezier curve was an unusual one in the history of applied mathematics. Most of the time when you find a real world problem needing a mathematical answer, you just find the math you need and apply it. Such is the case with Einstein's General Relativity and Riemann's Non-Euclidean Geometry discovered a century earlier.
Bezier worked as an engineer for a french automaker. To satisfy the needs of manufacturing, they needed a way of describing a curve exactly at every point. In those days, engineers sitting at drafting tables would would mark a starting point and an ending point of the curve they wanted, then pulled out a french curve and drew an approximate best-fit curve.
At the machine shop level, these best fit approximations were not good enough. In order for pieces to fit together the parts could only vary within certain tolerances, many of these approximate curves were outside the tolerances. By 1960, hardware became available that allowed the machining of 3D shapes out of blocks of wood or steel, known today as CAM or Computer Aided Manufacturing. Computer graphics was still in its infancy at the time, so designing a method of describing any curve you wanted was of utmost importance.
Bezier had to come up with a method of describing a best fit curve that would be easy to use and exact enough to meet the demands of manufacturing. Unfortunately, no mathematics existed at the time to do the job adequately(1). After numerous schemes, he came up with a method of describing any 2nd degree curve using only four points.
The method is rather simple. He starts by describing a curve inside a cube (the figure below to the left) using a parametric equation equal to the graph of y = x2. Then by transforming the cube into any kind of parallelepiped (below to the right), the curve will change as well. The four control points are the vertices of an imaginary parallelepiped. In the illustrations, points a and d represent the starting and ending points. Points b and c determine the curves depth and orientation. The slope of line ab is the starting slope of the curve, the slope of cd is the ending slope. Bezier's mathematical representation can be expanded to more than four control points to create curves of higher degrees, but for most uses four is enough.
For the real math junkies out there, the parametric function for Bezier Curve bn(t), where point A is b0, B is b1, etc. and n is the number of points - 1 is:
2nd degree Bezier Curves can be lined up one after another to create all kinds of shapes in two dimensions. But, what was really important to auto manufacturing was describing a whole piece in 3 dimensions. Putting four curves together in a square shape creates a bezier surface with 12 vertices, and creating tiles of these surfacescan create any three dimensional shape you can imagine.
In today's computer aided world, the applications are numerous. Not just in obvious applications like computer graphics and animation (animation often uses bezier curves applied to the fourth dimension to describe smooth motion), but also in robot controlled manufacturing. The Bezier Curve changed the world.
The Bezier Curve Easily Simplified
For you non-math savvy people out there, let me explain the math of the Bezier curve as easily as possible. We will use the most simplified case: a curve from three coordinate points. Think about "String Art". Start with three arbitrary points, A, B, and C. Draw a line AB and divide it evenly into 10 (or so) parts. Draw another line from B to C and divide it into 10 parts as well. Draw a line from A to B. Draw another from the point next to A to the point next to B. Continue until you go from B to C. The curve created is a close approximation of a Bezier Curve using three points. You could say Bezier created a method of describing the mathematics of String Art. 1. At least two mathematicians solved the problem before Bezier: Airplane designer James Ferguson, and engineer Paul de Casteljau who worked for Citroen. The latter's work is mathematically equivalent to Bezier, in fact the formula listed above is De Casteljau's. Unfortunately, their discoveries were closely guarded industrial secrets and were not published until after Bezier.http://members.cox.net/mathmistakes/bezier.htmSome images of String Art. |
