On the crowding of parameters associated with Schwarz-Christoffel transformations
type: Item
instance of: scholarly article
title: On the crowding of parameters associated with Schwarz-Christoffel transformations
zbMATH Open document ID: 0662.30007
DOI: 10.1016/0096-3003(88)90078-1
author: Basil C. Krikeles
author: Richard L. Rubin
published in: Applied Mathematics and Computation
publication date: 1988-01-01T00:00:00Z
review text: Let \(z_ 1,z_ 2,...,z_ n\) be n distinct points on \(\partial {\mathbb{D}}\) in counter clockwise order. Then the authors' problem is: For which linear map L of \({\mathbb{D}}\) onto itself are the images \(Z_ k=L(z_ k)\) least crowded, i.e. when is \(\min_{k}| Z_{k+1}-Z_ k|\) maximal \((Z_{n+1}=Z_ 1)?\) It is shown that the extremal L is uniquely determined, up to rotation, and that the \(Z_ k\) are least crowded if and only if for every semicircle \(S\subset \partial {\mathbb{D}}\) there is a pair \(Z_ k\), \(Z_{k+1}\) such that \[ | Z_{k+1}-Z_ k| =M:=\min_{k}| Z_{k+1}-Z_ k|, \] and at least one half of the arc from \(Z_ k\) to \(Z_{k+1}\) lies outside S. The problem is relevant in the use of the Schwarz-Christoffel formula; renormalization of the mapping can reduce the crowding phenomenon of the inverse images first noticed by the reviewer in Numer. Math. 19, 179-194 (1972; Zbl 0221.65053).
Mathematics Subject Classification ID: 30C30
zbMATH DE Number: 4081897
reviewed by: Dieter Gaier
describes a project that uses: SCPACK
MaRDI profile type: Publication
full work available at URL: 0096-3003(88)90078-1
OpenAlex ID: W2045893169
cites work: Konstruktive Methoden der konformen Abbildung
cites work: Calculation of Gauss Quadrature Rules
cites work: Q3699980
cites work: Q3285751
cites work: The Application of Numerical Methods to the Conformal Transformation of Polygonal Boundaries
cites work: Numerical conformal mapping via the Szegö kernel
cites work: Methods for numerical conformal mapping
cites work: Q3929965
cites work: Q5657604
cites work: Numerical Computation of the Schwarz–Christoffel Transformation
cites work: Determination of the parameters of the conformal mapping of simply connected polygonal regions
cites work: A conformal map formula for difficult cases
description: scientific article; zbMATH DE number 4081897