The second image (bottom left) shows the control mesh after one subdivision step. Each of the 24 facets of the control mesh has been split into 4 facets (for a total of 96 facets), and the points have been repositioned using a carefully constructed averaging process that Ed Calmull and Jim Clark created 20 years ago. (See "Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes." ComputerAidedDesign, 10(6), 1978.) In the third image (right) the mesh has been split and averaged once again, resulting in more facets.
The fourth image (bottom right) shows a final subdivision surface. The magic that guarantees that the final surface is smooth is tied up in the averaging process, most averaging methods create infinitely bumpy "tractal" surfaces. Recently, a rigorous mathematical proof of smoothness of the surface was constructed by Ulrich Reif (See "A Unitied Approach to Subdivision Algorithm," Technical Report A-92-16. Universitaet Stuttgart. 1992). As the math has become better, the software for creating smooth animated surfaces has become simpler.
Conceptually, the final surface is the infinitely faceted model that results if subdivision is repeated forever. Of course, in practice one cannot subdivide forever, so we've developed methods that allow highly accurate images to be created after just a few subdivision steps. Some of those methods are in a technical paper published in the SIGGHAPH 93 Conference Proceedings: "Efficient, Fair Interpolation Using Catmull-Clark Surfaces," by Mark Halstead, Michael Kass, and Tony DeRose. The new technology we've developed at Pixar since then is in the patents we've filed.