Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes
Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes
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平面三角形网格上的全纯向量场和二次微分
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
U. Pinkall
中科院分区：
文献类型：
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作者：
Wai Yeung Lam;U. Pinkall
给定复平面中的三角区域，离散向量场Y将向量（Y_iin mathbb {C}）分配给每个顶点。我们称这样一个向量场是全纯的，如果它定义了一个保持长交比的三角剖分的无穷小变形。我们表明，每个全纯向量场可以构造基于一个离散的调和函数的意义上的cotan拉普拉斯算子。此外，对于每一个全纯向量场，我们以莫比乌斯不变量的方式关联到某个全纯二次微分。在这里，二次微分被定义为一个对象，它为每个内部边缘分配一个纯虚数。然后我们推导了一个Weierstrass表示公式，它表明如何利用一个全纯二次微分来构造一个具有给定Gaus映射和给定Hopf微分的离散极小曲面.
Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector (Y_iin mathbb {C}) to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a Mobius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gaus map and prescribed Hopf differential.