Catmull–Clark subdivision surface

The Catmull-Clark algorithm is used in subdivision surface modeling to create smooth surfaces. It was devised by Edwin Catmull (of Pixar) and Jim Clark, and won an Academy Award for Technical Achievement in 2006.

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

• For each face, add a face point
  • Set each face point to be the centroid of all original points for the respective face.
  • 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 edge, add an edge point.
  • Set each edge point to be the average of all neighbouring face points and original points.
• For each original point P, take the average F of all n face points for faces touching P, and take the average R of all n edge midpoints for edges touching P, where each edge midpoint is the average of its two endpoint vertices. Move each original point to the point

${\displaystyle {F+2R+(n-3)P \over n}.}$

The new mesh will consist only of quadrilaterals, which won't in general be flat. 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 ${\displaystyle \mathbb {G} ^{1}}$ at extraordinary vertices and ${\displaystyle \mathbb {C} ^{2}}$ everywhere else.

• 3ds max
• Blender
• Gelato
• Cheetah3D
• Cinema4D
• LightWave 3D, version 9
• Maya
• modo
• Silo
• Softimage XSI
• Strata 3D CX
• Wings 3D

• E. Catmull and J. Clark: Recursively generated B-spline surfaces on arbitrary topological surfaces, Computer-Aided Design 10(6):350-355 (November 1978), (doi, pdf).
• Catmull-Clark Subdivision Surfaces
• Jos Stam: Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values, Computer Graphics Proceedings ACM SIGGRAPH 1998, 395-404, (pdf, downloadable eigenstructures)