Conformal Parameterization of Disks and Sphere Sectors

We prove a novel method for the embedding of a 3-fold rotationally symmetric sphere-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 image set being that its translations tile the plane, in turn this forces the angles at the embedding of the branch points in the construction. These parameterizations are optimal with respect to the Dirichlet energy functional defined on simplicial complexes, i.e. they are free boundary and harmonic. Since the parameterization is over a fixed area domain, it is conformal (i.e. a minimizer of the Least Squares Conformal Maps energy). The embedding is done by a novel construction of a torus from 63 copies of the original sphere. As a foundation for this result we first prove the optimality of the embedding of disk-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 be reduced by simpler constructions. 3-fold symmetric surfaces appear in nature, for example the surface of the 3-fold symmetric proteins PIEZO1 and PIEZO2 which are an important target of current studies (the 2021 Nobel Prize in Physiology or Medicine was awarded to researchers who studied the role of these proteins).