Mathematical Structures for Computer Graphics

7.5 Surfaces

The serious study of curves in computer graphics probably has many roots, but certainly Bézier curves began in automotive design where they served as silhouettes of the real goal, an aesthetically pleasing and aerodynamically efficient car body. Surfaces were the end result. Curves play several roles in computer graphics, and the two key uses are as paths for cameras and objects in animated sequences and as the skeletal elements in surface design. We can think of surfaces as some kind of mesh of curves.

Perhaps the most common surfaces expressed mathematically are the plane and the sphere . These implicit descriptions hide almost all hints of a curve, but intuition can prompt us to view the plane as a straight line swept in a constant direction and similarly the sphere as a circle rotated around a diameter.

One of the advantages of this viewpoint is that it can make it easier to design a surface and to find an array of points on the surface. Ultimately, in order to present a surface on the computer screen, we need an array of points. If the points are vertices in a triangular mesh, then rendering the surface is a matter of drawing many triangles, suitably shaded, on the screen. Finding the appropriate points is not a trivial task, but if we can visualize a collection ...