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 ∈ 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 Möbius 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 Gauß map and prescribed Hopf differential.

Minimal surfaces from infinitesimal deformations of circle packings

“…Motivated by his approach, we introduce vertex rotation to describe deformations of circle packings. Vertex rotation was first considered on circle patterns where neighboring circles intersect [17].…”

“…Instead of circles, vertex scaling concerns the edge lengths of the triangle mesh [18,4]. Both theories are closely related and can be expressed neatly in terms of complex cross ratios [17].…”

Isothermic triangulated surfaces

“…The choice of such a pair of holomorphic functions is equivalent to prescribing the Gauss map and the Hopf differential of a minimal surface. A similar formula for discrete minimal surfaces is elaborated in [28], where a notion of holomorphic quadratic differentials is introduced to link discrete harmonic functions and discrete minimal surfaces by considering infinitesimal conformal deformations. Furthermore, it turns out that our definition of discrete minimal surfaces bridges the gap between earlier notions of discrete minimal surfaces [27].…”

Holomorphic quadratic differentials on graphs and the chromatic polynomial

“…Holomorphic quadratic differentials on surface graphs arise in discrete differential geometry, see [7] and Section 2 below. In some situations we drop condition (1); see below.…”

“…Holomorphic quadratic differentials have a surprising number of geometric applications. They were introduced in [7] for cell decompositions of surfaces. For a triangulated disk in the plane, it was shown there that there is a one-to-one correspondence between holomorphic quadratic differentials, infinitesimal (discrete) conformal deformations, discrete harmonic functions for the cotangent Laplacian and discrete minimal surfaces.…”