Spherical Orbifold Tu e Embeddings

This work presents an algorithm for injectively parameterizing surfaces into spherical target domains called spherical orbifolds. Spherical orbifolds are cone surfaces that are generated from symmetry groups of the sphere. The surface is mapped the spherical orbifold via an extension of Tutte’s embedding. This embedding is proven to be bijective under mild additional assumptions, which hold in all experiments performed. This work also completes the adaptation of Tutte’s embedding to orbifolds of the three classic geometries – Euclidean, hyperbolic and spherical – where the rst two were recently addressed. The spherical orbifold embeddings approximate conformal maps and require relatively low computational times. The constant positive curvature of the spherical orbifolds, along with the exibility of their cone angles, enables producing embeddings with lower isometric distortion compared to their Euclidean counterparts, a fact that makes spherical orbifolds a natural candidate for surface parameterization.