In 2005/06, Edwin Catmull, together with Tony DeRose and Jos Stam, received an Academy Award for Technical Achievement for their invention and application of subdivision surfaces. DeRose wrote about "efficient, fair interpolation" and character animation. Stam described a technique for a direct evaluation of the limit surface without recursion.
Set each edge point to be the average of the two neighbouring face points (A,F) and the two endpoints of the edge (M,E)[2]File:Catmull-Clark Recursive Step 2.pngEdge points (magenta cubes)
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). (Note that from the perspective of a vertex P, the number of edges neighboring P is also the number of adjacent faces, hence n)
Move each original point to the new vertex point (This is the barycenter of P, R and F with respective weights (n − 3), 2 and 1)File:Catmull-Clark Recursive Step 3.pngNew vertex points (green cones)
Form edges and faces in the new mesh
Connect each new face point to the new edge points of all original edges defining the original faceFile:Catmull-Clark Recursive Step 4.pngNew edges, 4 per face point
Connect each new vertex point to the new edge points of all original edges incident on the original vertex File:Catmull-Clark Recursive Step 5.png3 new edges per vertex point of shifted original vertices
The new mesh will consist only of quadrilaterals, which in general will not be planar. The new mesh will generally look "smoother" (i.e. less "jagged" or "pointy") than the old mesh. Repeated subdivision results in meshes that are more and more rounded.
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 they do go to great lengths to rigorously show that the method converges to bicubic B-spline surfaces.[1]
It can be shown that the limit surface obtained by this refinement process is at least at extraordinary vertices and everywhere else (when n indicates how many derivatives are continuous, we speak of continuity). After one iteration, the number of extraordinary points on the surface remains constant.
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 (1998).[3] This method reformulates the recursive refinement process into a matrix exponential problem, which can be solved directly by means of matrix diagonalization.
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