圆盘与球面扇区的快速共形参数化
Fast Conformal Parameterization of Disks and Sphere Sectors
AI总结：
本文提出一种快速共形参数化方法，通过63个球面副本构造环面，将3重旋转对称球面网格最优嵌入到具有相同对称性的平面区域，并证明了圆盘型网格嵌入特殊三角形和矩形的最优性。
摘要：
我们证明了一种新方法，用于将具有3重旋转对称性的球面型网格嵌入到具有3重旋转对称性的平面子集上。该嵌入是自由边界的，对图像集的唯一额外约束是其平移能够铺满整个平面，这反过来又迫使构造中分支点嵌入处的角度被确定下来。这些参数化关于定义在单纯复形上的Dirichlet能量泛函是最优的。由于参数化是在固定面积域上进行的，因此它是共形的（即LSCM能量的极小化子）。该嵌入通过一种新颖的构造实现，即由原始球面的63个副本构造一个环面。作为该结果的基础，我们首先证明了圆盘型网格嵌入到平面中特殊类型三角形和矩形上的最优性。3重对称环面的嵌入是满秩的，因此不能通过更简单的构造来约化。3重对称曲面在自然界中出现，例如3重对称蛋白PIEZO1和PIEZO2的表面，它们是当前研究的重要目标。
英文摘要：
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 is 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. Since the parameterization is over a fixed area domain, it is conformal (i.e. a minimizer of the LSCM 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
