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Main Title: Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization
Author(s): Bobenko, Alexander I.
Sechelmann, Stefan
Springborn, Boris
Type: Book Part
Language Code: en
Is Part Of: 10.1007/978-3-662-50447-5
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}, as hyperelliptic curves, and as \mathbb {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.
Issue Date: 2016
Date Available: 1-Sep-2017
DDC Class: 510 Mathematik
Book Title: Advances in discrete differential geometry
Editor: Bobenko, Alexander I.
Publisher: Springer
Publisher Place: Berlin, Heidelberg
Publisher DOI: 10.1007/978-3-662-50447-5_1
Page Start: 1
Page End: 56
ISBN: 978-3-662-50447-5
Appears in Collections:FG Geometrie und Integrable Systeme » Publications

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