Catmull–Clark subdivision surface

The Catmull–Clark algorithm is a technique used in 3D computer graphics to create curved surfaces by using subdivision surface modeling. It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology.

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.

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 (A,F) and the two endpoints of the edge (M,E)

A

+

F

+

M

+

E

4

{\displaystyle {\frac {A+F+M+E}{4}}}

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).