Catmull–Clark subdivision surface

Recursive evaluation

Catmull–Clark surfaces are defined recursively, using the following refinement scheme:

Start with a mesh of an arbitrary polyhedron. All the vertices in this mesh shall be called original points.

For each face, add a face point

Set each face point to be the average of all original points for the respective face.

For each edge, add an edge point.

Set each edge point to be the average of the two neighbouring face points and its two original endpoints.

For each face point, add an edge for every edge of the face, connecting the face point to each edge point for the face.

For each original point P, take the average F of all n (recently created) face points for faces touching P, and take the average R of all n edge midpoints for (original) edges touching P, where each edge midpoint is the average of its two endpoint vertices (not to be confused with new "edge points" above). Move each original point to the point

This is the barycenter of P, R and F with respective weights (n − 3), 2 and 1.

Connect each new vertex point to the new edge points of all original edges incident on the original vertex.

Define new faces as enclosed by edges.

The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look smoother than the old mesh.

Repeated subdivision results in smoother meshes. It can be shown that the limit surface obtained by this refinement process is at least C 1 at extraordinary vertices and C 2 everywhere else (when n indicates how many derivatives are continuous, we speak of C n continuity). After one iteration, the number of extraordinary points on the surface remains constant.

The arbitrary-looking barycenter formula was chosen by Catmull and Clark based on the aesthetic appearance of the resulting surfaces rather than on a mathematical derivation, although Catmull and Clark do go to great lengths to rigorously show that the method converges to bicubic B-spline surfaces.

Exact evaluation

The limit surface of Catmull–Clark subdivision surfaces can also be evaluated directly, without any recursive refinement. This can be accomplished by means of the technique of Jos Stam. This method reformulates the recursive refinement process into a matrix exponential problem, which can be solved directly by means of matrix diagonalization.