Fast Dirichlet Optimal Parameterization of Disks and Sphere Sectors

We utilize symmetries of tori constructed from copies of given disk-type meshes in 3d, together with symmetries of corresponding tilings of fundamental domains of plane tori. We use these correspondences to prove optimality of the embedding of the mesh onto special types of triangles in the plane, and rectangles. The proof provides a certain framework for using symmetries of the image domain. The complexity is linear in the mesh size. We then use the method to prove a novel embedding of a 3-fold rotationally symmetric sphere-type mesh onto a set in the plane with 3-fold rotational symmetry. The only additional constraint on the set is that its translations tile the plane. The embedding is optimal under the symmetry and tiling constraint. This is done by a novel construction of a torus from 63 copies of the original sphere.