Seamless Parametrization of Spheres with Controlled Singularities

Abstract

We present a method for constructing seamless parametrization for genus‐0 surfaces, which can handle any feasible cone configuration, thus allowing users to arbitrarily design and tailor a mapping as desired. The method directly constructs a self‐overlapping metapolygon of the domain boundary of the mapped cut mesh, which relieves the need of using an auxiliary surface. This simplifies the pipeline and allows for a necessary optimization of the boundary polygon before mapping the interior. Moreover, it enables handling larger meshes with more cones than previous methods can handle. Our construction is purely combinatorial, and it guarantees that the mapping is locally injective – a prerequisite to today's advanced optimization methods. This is achieved via careful construction of a simple domain boundary polygon, where existence of such a polygon is proven for all cases. We offer a numerically robust algorithm to automate the construction, which involves a solution of two linear problems. We offer a full pipeline, suggesting elegant solutions to sub‐problems, and demonstrate robustness through extensive experiments.