Constructing conformal maps of triangulated surfaces
type: Item
instance of: scholarly article
title: Constructing conformal maps of triangulated surfaces
zbMATH Open document ID: 1242.30007
author: G. Brock Williams
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-03-29T00:00:00Z
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 52C26
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 6019410
zbMATH Keywords: circle packing
zbMATH Keywords: conformal maps
zbMATH Keywords: conformal welding
zbMATH Keywords: triangulated surfaces
zbMATH Keywords: conformal brain flattening
reviewed by: David A. Brannan
describes a project that uses: CirclePack
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.01.023
OpenAlex ID: W2090585748
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DOI: 10.1016/J.JMAA.2012.01.023
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review text: The author describes a means of computing the uniformizing conformal map of a triangulated surface whose triangles are realized as Euclidean triangles in \(\mathbb{R}^3\) onto a fundamental domain in the unit disc \(\mathbb{D}\), plane \(\mathbb{C}\) or sphere \(S^2\); that such a mapping exists and is unique up to Möbius transformations was proved by \textit{A. F. Beardon} and \textit{K. Stephenson} [Indiana Univ. Math. J. 39, No. 4, 1383--1425 (1990; Zbl 0797.30008)].NEWLINENEWLINE His approach uses a circle packing technique (see, for example, [\textit{K. Stephenson}, Introduction to circle packing. The theory of discrete analytic functions. Cambridge: Cambridge University Press (2005; Zbl 1074.52008)]) to create a quasiconformal approximation to the conformal map, and then a discrete form of conformal welding to reduce the distortion and converge in the limit to the required conformal map.NEWLINENEWLINE Mappings of triangulated surfaces arise in areas such as conformal brain flattening (see, for example, [\textit{K. Stephenson}, Notices Am. Math. Soc. 50, No. 11, 1376--1388 (2003; Zbl 1047.52016)]).
description: scientific article; zbMATH DE number 6019410