A discrete curve-shortening equation
type: Item
instance of: scholarly article
title: A discrete curve-shortening equation
zbMATH Open document ID: 0897.34010
author: Kazuaki Nakayama
author: Harvey Segur
author: Miki Wadati
published in: Methods and Applications of Analysis
publication date: 1998-10-06T00:00:00Z
review text: The authors propose a new discrete version of the so-called curve-shortening equation \(d{\mathbf r}/dt=\kappa{\mathbf n}\) where \(\kappa\) denotes the curvature and \({\mathbf n}\) the inner unit normal vector. It is chosen so that it inherits the following geometric feature of the continuous equation: the area \(S\) enclosed by the curve satisfies \(dS/dt=-2\pi\) which implies that every closed curve shrinks to a point in finite time. In the continuous case one can show that the asymptotic shape of such a shrinking curve is a circle. The authors conjecture that for their discrete version a similar property holds, namely that its asymptotic shape is an equiangular \(n\)-polygon. They treat in detail the case of a triangle, a parallelogram and of a hexagon.
Mathematics Subject Classification ID: 34A26
Mathematics Subject Classification ID: 74A15
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 1129314
zbMATH Keywords: curve-shortening equation
zbMATH Keywords: discretization
zbMATH Keywords: geometric model
reviewed by: Werner M. Seiler
MaRDI profile type: Publication
DOI: 10.4310/MAA.1997.V4.N2.A6
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description: scientific article; zbMATH DE number 1129314