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A. Vaxman, C. Müller, O. Weber:
"Canonical Möbius Subdivision";
in: "Proc. Siggraph Asia 2018 Technical Papers", 37 (6) Article 227; ACM SIGGRAPH, New York City, New York, 2018, ISBN: 978-1-4503-6008-1.

We present a novel framework for creating Möbius-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 Möbius geometry; namely, reproducing spheres,circular arcs, and Möbius 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 Möbius 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 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. 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.

http://dx.doi.org/10.1145/3272127.3275007

http://dmg.tuwien.ac.at/geom/ig/publications/moebiussubdivision/moebiussubdivision.pdf