Orbifold Tutte embeddings

Injective parameterizations of surface meshes are vital for many applications in Computer Graphics, Geometry Processing and related fields. Tutte's embedding, and its generalization to convex combination maps, are among the most popular approaches for computing parameterizations of surface meshes into the plane, as they guarantee injectivity, and their computation only requires solving a sparse linear system. However, they are only applicable to disk-type and toric surface meshes. In this paper we suggest a generalization of Tutte's embedding to other surface topologies, and in particular the common, yet untreated case, of sphere-type surfaces. The basic idea is to enforce certain boundary conditions on the parameterization so as to achieve a Euclidean orbifold structure. The orbifold-Tutte embedding is a seamless, globally bijective parameterization that, similarly to the classic Tutte embedding, only requires solving a sparse linear system for its computation. In case the cotangent weights are used, the orbifold-Tutte embedding globally minimizes the Dirichlet energy and is shown to approximate conformal and four-point quasiconformal mappings. As far as we are aware, this is the first fully-linear method that produces bijective approximations to conformal mappings. Aside from parameterizations, the orbifold-Tutte embedding can be used to generate bijective inter-surface mappings with three or four landmarks and symmetric patterns on sphere-type surfaces.