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.

中文翻译：

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圆盘和球扇区的快速狄利克雷最优参数化

我们利用由 3d 中给定磁盘类型网格的副本构建的环面对称性，以及平面环面基本域的相应平铺的对称性。我们使用这些对应关系来证明网格嵌入平面中特殊类型的三角形和矩形的最优性。该证明为使用图像域的对称性提供了一定的框架。复杂性与网格大小呈线性关系。然后，我们使用该方法证明了将 3 重旋转对称球型网格嵌入到具有 3 重旋转对称平面的集合上的新方法。对集合唯一的额外约束是它的平移平铺了平面。在对称和平铺约束下嵌入是最优的。这是通过从原始球体的 63 个副本中构建的一种新颖的圆环体来完成的。