A novel framework for creating Möbius-invariant subdivision operators with a simple conversion of existing linear subdivision operators is presented. With this, a wide variety of subdivision surfaces that have properties derived from Möbius geometry; namely, reproducing spheres, circular arcs, and Möbius regularity, are obtained. The respective method is based on establishing a canonical form for each 1-ring in the mesh, representing the class of all 1-rings that are Möbius equivalent to that 1-ring. This is done by performing a chosen linear subdivision operation on these canonical forms, and blending the positions contributed from adjacent 1-rings, using two novel Möbius-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. The 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.
Amir Vaxman, Christian Müller, and Ofir Weber.
Canonical Möbius Subdivision.
ACM Trans. Graphics (Proc. SIGGRAPH ASIA), 2018.