On Toy Story, smooth characters were a patchwork, and Pixar's own paper says "considerable manual effort was required to hide the seams in the face of Woody." A patch is a rectangle, a face isn't, and as the model moves the seams want to open. For the short Geri's Game (1997), Tony DeRose, Michael Kass and Tien Truong started from a rough cage instead: "The control mesh for Geri's head, created by digitizing a full-scale model sculpted out of clay." Then Ed Catmull and Jim Clark's rule from 1978: a new point in every face and every edge, each old corner pulled toward its neighbours, again and again, until the cage is skin. For fingernails, each crease got a sharpness: how many rounds it keeps the sharp rule before the smooth one takes over. Geri's skin, hands, jacket, shirt and tie were all made this way, and in the paper's words "smoothness of the model is automatically guaranteed, even as the model animates."

Our own clay head, cage and fingertip, computed here; no frames from any film.

What would you build from a rough cage first?

Sources: Tony DeRose, Michael Kass, Tien Truong, "Subdivision Surfaces in Character Animation", SIGGRAPH 98 (Pixar); Ed Catmull and Jim Clark, "Recursively generated B-spline surfaces on arbitrary topological meshes", Computer-Aided Design, 1978.
Sounds: our own, made in code (earshot, CC0).