论文标题
磁盘和球体的快速共形参数化
Fast Conformal Parameterization of Disks and Sphere Sectors
论文作者
论文摘要
我们证明了一种新的方法，可以将3倍旋转的对称球体型网嵌入具有3倍旋转对称性的平面子集中。嵌入是自由边缘的，对图像集的唯一额外约束是其平面瓷砖，而这又迫使构造中分支点嵌入的角度。这些参数化相对于在简单复合物上定义的dirichlet能量功能是最佳的。由于参数化在固定区域域上，因此是共形的（即LSCM能量的最小化器）。嵌入是通过从63份原始球体副本中的圆环的新颖结构来完成的。作为此结果的基础，我们首先证明了将磁盘类型嵌入到平面中特殊类型的三角形和矩形上的最佳性。 3倍对称圆环的嵌入是完整的等级，因此不能通过更简单的构造来减少。 3倍对称表面出现在自然界中，例如3倍对称蛋白质压电和压电2的表面，这是当前研究的重要目标。
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.
