Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector Yi ∈ 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.
CITATION STYLE
Lam, W. Y., & Pinkall, U. (2016). Holomorphic vector fields and quadratic differentials on planar triangular meshes. In Advances in Discrete Differential Geometry (pp. 241–265). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-50447-5_7
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