Math Shapes: Geri
Tony DeRose is one of the major designers responsible for the character Geri, who appears in the Academy Award-winning short film "Geri's Game" as well as in "Toy Story II". He is well known for his presentations about computer graphics and the steps that people have to go through to learn about careers in that subject. Another person at Pixar with a keen interest in outreach as well as major responsibility for the production of Pixar animations is the CTO, Edwin Catmull.
Computer animation depends on mathematics. Mathematical modeling techniques develop all of the characters in an animated film, and all of the positions and movements have explicit mathematical descriptions. It is impossible to think of modern computer graphics without at the same time thinking of the mathematics that underlies it. A Simple Java applet will let you experiment with these ideas.
To create Geri, Pixar Animation Studios used a fairly novel surface modeling method known as subdivision surfaces. The idea behind subdivision surfaces is that a smooth surface is from a fairly coarse polyhedron by iteractively refining — that is, subdividing — the polygonal facets. If your refinement rules are constructed properly you can show that in the limit you converge to a smooth surface. Geri's head was constructed with a single subdivision surface where the coarse polyhedron contained about 4500 vertices.
Computer Graphics World wrote an article on the construction of Geri, indicating many of the mathematical ideas involved in this kind of work. (It is possible to enroll at this site and see the text and some of the pictures in the article, without agreeing to subscribe or to receive e-mail solicitations from the magazine.)
Here is a description of the subdivision surface process, using illustrations from the original of the above article which are not included in the online version.
There is an official Geri web site, hosted by Pixar Animation Studios, that includes a short clip from "Geri's Game". Related sites are the Internet Movie Database entry for Geri and Pixar's Shorts.