Catmull Clark Subdivision

The only difference between Catmull-Clark and linear subdivision is the choice of positions for new vertices. Whereas linear subdivision simply takes a uniform average of the old vertex positions, Catmull-Clark uses a very carefully-designed weighted average to ensure that the surface converges to a nice, round surface as the number of subdivision steps increases. The original scheme is described in the paper "Recursively generated B-spline surfaces on arbitrary topological meshes" by (Pixar co-founder) Ed Catmull and James Clark. Since then, the scheme has been thoroughly discussed, extended, and analyzed; more modern descriptions of the algorithm may be easier to read, including those from the Wikipedia and this webpage. In short, the new vertex positions can be calculated by:

1. setting the new vertex position at each face f to the average of all its original vertices (exactly as in linear subdivision),
2. setting the new vertex position at each edge e to the average of the new face positions (from step 1) and the original endpoint positions, and
3. setting the new vertex position at each vertex v to the weighted sum

where n is the degree of vertex v (i.e., the number of faces containing v), and

• Q is the average of all new face position for faces containing v,
• R is the average of all original edge midpoints for edges containing v, and
• S is the original vertex position for vertex v.

In other words, the new vertex positions are an "average of averages." (Note that you will need to divide by n both when computing Q and R, and when computing the final, weighted value---this is not a typo!)

Apart from changing the way vertex positions are computed, there should be no difference in your implementation of linear and Catmull-Clark subdivision.

This step should be implemented in the method HalfedgeMesh::computeCatmullClarkPositions in meshEdit.cpp.

This subdivision rule is not required to support meshes with boundary, unless the implementer wishes to go above and beyond.