Vertices and normals passing trough a point of convex curves of constant width and hedgehogs singularities
type: Item
instance of: scholarly article
title: Vertices and normals passing trough a point of convex curves of constant width and hedgehogs singularities
zbMATH Open document ID: 1025.52004
DOI: 10.1007/BF02638386
author: Yves Martinez-Maure
published in: Archiv der Mathematik
publication date: 2003-03-27T00:00:00Z
review text: It is known that planar convex curves of constant width, of class \(C^4\) and with strictly positive curvature have at least 6 vertices [\textit{G. D. Chakerian} and \textit{H. Groemer}, Convexity and its applications, Collect. Surv., 49-96 (1983; Zbl 0518.52002)]. It is also known that these curves either contain a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals [\textit{Y. Martinez-Maure}, Publ. Mat., Barc. 44, 237-255 (2000; Zbl 0974.53003)]. The author proves in this paper that both properties are closely related. The main result states that if with the above assumptions all its vertices are nondegenerate, then (i) the curve has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of which pass at least 6 normals; (ii) if the curve has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof: (i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates this index to the number of singularities of the evolute (i.e. of vertices). Besides the author gives some formulae for counting singularities of generic hedgehogs in euclidean space of dimensions 2 and 3.
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 1889279
zbMATH Keywords: curves of constant width
zbMATH Keywords: vertices
zbMATH Keywords: normals
zbMATH Keywords: singularities
zbMATH Keywords: hedgehog
zbMATH Keywords: evolute
zbMATH Keywords: index
zbMATH Keywords: Gauss-Bonnet formula
zbMATH Keywords: swallowtail
zbMATH Keywords: convex curves
zbMATH Keywords: support function
zbMATH Keywords: geodesic curvature
MaRDI profile type: Publication
cites work: Q3914103
cites work: Korrektur zu: Existenz eines 6-Normalenpunktes in einem konvexen Körper
cites work: On projective hedgehogs (envelopes parametrized by their Gauss map)
cites work: Hedgehog index: Study and applications
full work available at URL: https://doi.org/10.1007/bf02638386
OpenAlex ID: W1987431656
description: scientific article; zbMATH DE number 1889279