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We prove a novel method for the embedding of a 3-fold rotationally symmetricsphere-type mesh onto a subset of the plane with 3-fold rotational symmetry.The embedding is free-boundary with the only additional constraint on the imageset is that its translations tile the plane, in turn this forces the angles atthe embedding of the branch points in the construction. These parameterizationsare optimal with respect to the Dirichlet energy functional defined onsimplicial complexes. Since the parameterization is over a fixed area domain,it is conformal (i.e. a minimizer of the LSCM energy). The embedding is done bya novel construction of a torus from 63 copies of the original sphere. As afoundation for this result we first prove the optimality of the embedding ofdisk-type meshes onto special types of triangles in the plane, and rectangles.The embedding of the 3-fold symmetric torus is full rank and so cannot bereduced by simpler constructions. 3-fold symmetric surfaces appear in nature,for example the surface of the 3-fold symmetric proteins PIEZO1 and PIEZO2which are an important target of current studies.

Fast Conformal Parameterization of Disks and Sphere Sectors