Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

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 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.