A subdivision surface is a surface defined as the limit of repeated refinement of a polygonal surface mesh. A subdivision rule specifies both the connectivity of the refined mesh and the positions of its vertices. Starting with a finite control mesh, the rule is applied successively to produce finer control meshes whose limit, when it exists, is the subdivision surface (Catmull and Clark 1978, DeRose et al. 1998).
In a linear subdivision scheme, each new vertex position is a weighted sum of old vertex positions. When the weights have sum one, refinement commutes with affine transformations. The control meshes and the limiting surface are distinct objects, and the limiting surface need not pass through the original control vertices. Convergence and smoothness depend on the chosen refinement rules and the local mesh connectivity (DeRose et al. 1998, Numberphile 2014).
Catmull-Clark subdivision is a construction that generalizes uniform cubic B-spline refinement to polygonal control meshes with arbitrary topology. Its face, edge, and vertex rules replace each old face by quadrilaterals. Regular quadrilateral regions use the tensor-product cubic B-spline rules (Catmull and Clark 1978).
