Canonical Mobius Subdivision

We present a novel framework for creating Mobius-invariant subdivision operators with a simple conversion of existing linear subdivision operators. By doing so, we create a wide variety of subdivision surfaces that have properties derived from Mbbius geometry; namely, reproducing spheres, circular arcs, and Mobius regularity. Our method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Mbbius equivalent to that 1-ring. We perform a chosen linear subdivision operation on these canonical forms, and blend the positions contributed from adjacent 1-rings, using two novel Mobius-invariant operators, into new face and edge points. The generality of the method allows for easy coarse-to-fine mesh editing with diverse polygonal patterns, and with exact reproduction of circular and spherical features. Our operators are in closed-form and their computation is as local as the computation of the linear operators they correspond to, allowing for efficient subdivision mesh editing and optimization.