Minimal enclosing discs, circumcircles, and circumcenters in normed planes. I.
type: Item
instance of: scholarly article
title: Minimal enclosing discs, circumcircles, and circumcenters in normed planes. I.
zbMATH Open document ID: 1245.65022
author: Javier Alonso
author: Horst Martini
author: Margarita Spirova
published in: Computational Geometry
publication date: 2012-05-18T00:00:00Z
reviewed by: Jason Hanson
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 51B20
zbMATH DE Number: 6036385
zbMATH Keywords: circumcenters
zbMATH Keywords: intersection of norm circles
zbMATH Keywords: minimal enclosing balls
zbMATH Keywords: Minkowski geometry
zbMATH Keywords: normed plane
MaRDI profile type: Publication
full work available at URL: j.comgeo.2012.01.007
OpenAlex ID: W2038986281
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DOI: 10.1016/J.COMGEO.2012.01.007
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review text: Let \(\|\cdot\|\) be a norm for the vector space \({\mathbb R}^2\), and suppose we are given three non--colinear points \(t_1,t_2,t_3\) in the plane. In contrast to the Euclidean case, the intersection of two circles may contain line segments, and there may be multiple circumcircles that contain the points \(t_1,t_2,t_3\). In this article, the authors classify all possible intersections of two circles in the plane. This classification is then used to explicitly determine the regions in the plane that consist of points \(x\) for which there exists a norm such that \(x\) is the center of a circumcircle of \(t_1,t_2,t_3\). Moreover for the given norm \(\|\cdot\|\), in the case where the line segment from \(t_1\) to \(t_2\) lies in the intersection of two distinct circumcircles \({\mathcal C},{\mathcal C}'\) with centers \(c,c'\), the authors determine further restrictions on the regions where the centers can be located; in particular, it is shown that \({\mathcal C},{\mathcal C}'\) have the same radius if and only if \(c,c'\) lie on the line passing through the midpoint of \(t_1\) and \(t_3\) and the midpoint of \(t_2\) and \(t_3\).NEWLINENEWLINEThe article is largely self-contained and intended for a general mathematical audience.
description: scientific article; zbMATH DE number 6036385