CatmullClark

Catmull-Clark is a subdivision scheme used in computer graphics to generate smooth surfaces from coarse polygonal meshes. It was introduced by Ed Catmull and Jim Clark in 1978 and has become a foundational technique in 3D modeling and animation, notably influencing Pixar’s subdivision workflows.

In each subdivision step, the algorithm computes three kinds of points: face points, edge points, and new vertex positions. For every polygonal face, a face point is created as the average of the face’s vertices. For every edge, an edge point is created as the average of the edge’s endpoints and the face points of the two adjacent faces. Each original vertex is repositioned to a new location based on the surrounding face points, edge points, and the vertex’s valence. After all new points are computed, the original face is replaced by smaller faces that connect the new points, typically producing a mesh with more quad faces. Repeating the process yields a progressively smoother limit surface. The method works with meshes composed of polygons of any size, though it tends to produce quad-dominant geometry.

Catmull-Clark surfaces are generally smooth in the limit, with high continuity at regular regions and slightly reduced continuity at irregular vertices (where the valence differs from four). The scheme is widely adopted for modeling and animation, and many other subdivision schemes build on its concepts or provide alternative refinement rules.