Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization

Alexander I. Bobenko (), Stefan Sechelmann () and Boris Springborn ()
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Alexander I. Bobenko: Technische Universität Berlin, Inst. für Mathematik
Stefan Sechelmann: Technische Universität Berlin, Inst. für Mathematik
Boris Springborn: Technische Universität Berlin, Inst. für Mathematik

A chapter in Advances in Discrete Differential Geometry, 2016, pp 1-56 from Springer

Abstract: Abstract We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in $$\mathbb {R}^{3}$$ R 3 , as hyperelliptic curves, and as $$\mathbb {CP}^{1}$$ CP 1 modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller.

Date: 2016
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DOI: 10.1007/978-3-662-50447-5_1

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