How To Draw a Sphere-Part 1

nce upon a time (before the invention of ray trac0 ing) I could claim the honor of having drawn more spheres than anybody in the world. This was because of all the moons and planets I drew for the Voyager flyby movies at the Jet Propulsion Lab. I could have drawn spheres by hacking them up into scads of polygons, but that would have been the coward's way out. Instead I wrote a special-purpose program that was highly optimized for drawing spheres. This program involved quite a variety of interesting tricks, but when I started out to write about them, I realized that a whole bunch of matrix mathematics background is necessary to understand the sphere-rendering algorithm. This column, then, is largely the matrix mathematics necessary to manipulate (and ultimately render) second-order surfaces. I'll get down to the rendering next time.