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arXiv · 1506.08099

Holomorphic vector fields and quadratic differentials on planar triangular meshes

Wai Yeung Lam·Ulrich Pinkall
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Abstract

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ 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 M\"obius 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 Gau{\ss} map and prescribed Hopf differential.

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Wai Yeung Lam, Ulrich Pinkall. 2015-06-26. Holomorphic vector fields and quadratic differentials on planar triangular meshes. https://arxiv.org/abs/1506.08099

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