# Catmull-Clark Subdivision

For an in-practice example, see the User Guide.

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 adjacent face positions (from step 1) and the original edge endpoint positions, and
3. setting the new vertex position at each vertex $$v$$ to the weighted sum
$\frac{Q+2R+(n-3)S}{n}$

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!)

Your implementation of linear and Catmull-Clark subdivision will be very similar - only differing on how to compute the vertices new positions at each edge and vertex.

This step should be implemented in the method HalfedgeMesh::catmullclark_subdivide_positions in student/meshedit.cpp.

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