Crofton formulas and convexity condition for secantoptics
type: Item
instance of: scholarly article
title: Crofton formulas and convexity condition for secantoptics
zbMATH Open document ID: 1178.53001
author: Witold Mozgawa
author: Magdalena Skrzypiec
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2009-10-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1251832370
review text: The paper under review deals with some properties of secantoptics, defined in [\textit{M. Skrzypiec}, Beitr. Algebra Geom. 49, No.~1, 205--215 (2008; Zbl 1155.53003)]. An isoptic of a given curve \(C\) is a set of points from which this curve is seen under the fixed angle \(\alpha \in (0,\pi)\). A hedgehog is a curve which can be parametrized by the formula \(z(t)=\psi(t)e^{it}+\dot\psi(t)ie^{it}\) with \(\psi(t)=h(\cos t,\sin t), h\in C^2(\mathbb S^1,\mathbb R)\). A closed convex curve \(C\) of class \(C^2\) with nonvanishing curvature is said to be an oval. The evolutoid of angle \(\delta\) of a curve \(f(s)\) is the envelope of the lines making a fixed angle \(\delta\) with the normal vector at \(f(s)\). The authors show that any evolutoid of a given oval \(C\) is a hedgehog and that any secantoptic of an oval \(C\), is an isoptic of a pair of required evolutoids. They also prove some Crofton-type formulas for secantoptics and give a necessary and sufficient condition for a secantoptic to be convex.
reviewed by: Valeriĭ V. Volchkov
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 5612289
zbMATH Keywords: secantoptic
zbMATH Keywords: isoptic
zbMATH Keywords: secant
zbMATH Keywords: hedgehog
zbMATH Keywords: evolutoid
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5612289