Catmull–Clark subdivision surface

The Catmull–Clark algorithm is a technique used in computer graphics to create smooth surfaces by 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, Edwin Catmull received an Academy Award for Technical Achievement together with Tony DeRose and Jos Stam for their invention and application of subdivision surfaces.

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.

The new mesh will consist only of quadrilaterals, which in general will not be planar. 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 ${\displaystyle {\mathcal {C}}^{1}}$ at extraordinary vertices and ${\displaystyle {\mathcal {C}}^{2}}$ everywhere else (when n indicates how many derivatives are continuous, we speak of ${\displaystyle {\mathcal {C}}^{n}}$ continuity). After one iteration, the number of extraordinary points on the surface remains constant.

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