Patching Catmull-Clark Meshes
Siggraph 2000
Jörg Peters

 A simple, explicit transformation creates maximally large, smoothly joining Nurbs patches of degree(3,3) from Catmull-Clark subdivision meshes. It can be applied after any of the first subdivision steps and creates patches that are maximally large in the sense that one patch corresponds to one quadrilateral facet of the initial, coarsest quadrilateral mesh before subdivision. The patches join C^2 almost everywhere and with tangent continuity in the immediate neighborhood of extraordinary mesh nodes, matching the global smoothness of Catmull-Clark limit surfaces. The transitions between patches are almost all parametric. Named after the title, the PCCM transformation integrates naturally with array-based implementations of subdivision surfaces. See the Siggraph talk slides for remarks on choice of normal, even higher-order saddle points and blend ratios and smoothed creases.

Algorithm implemented and figures rendered by  Le-Jeng Shiue.
PCCM surface and 
input mesh
PCCM surface with 
improved  rule of higher-order saddle point 
PCCM surface with
the rule of higher-order saddle point  in the SIGGRAPH paper
PCCM surface with 
blend ratio of .80