Catmull–Clark subdivision surface

It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology.

[1] 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.

Stam described a technique for a direct evaluation of the limit surface without recursion.

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.

After one iteration, the number of extraordinary points on the surface remains constant.

Catmull–Clark level-3 subdivision of a cube with the limit subdivision surface shown below. (Note that although it looks like the bi-cubic interpolation approaches a sphere , an actual sphere is quadric .)
Visual difference between sphere (green) and Catmull-Clark subdivision surface (magenta) from a cube
Face points (blue spheres)
Edge points (magenta cubes)
New vertex points (green cones)
New edges, 4 per face point
3 new edges per vertex point of shifted original vertices
Final faces to the mesh