On evolutoids of planar convex curves
type: Item
instance of: scholarly article
title: On evolutoids of planar convex curves
zbMATH Open document ID: 1307.53004
author: D. Kharzeev
published in: Aequationes Mathematicae
publication date: 2014-10-02T00:00:00Z
review text: Let \(\gamma\) be a (regular) planar closed convex curve and consider a fixed angle \(\alpha\). The evolutoid \(\gamma_{\alpha}\) of \(\gamma\) is the envelope of lines making an angle \(\alpha\) with the tangent lines at \(\gamma\). Denote by \(A(\gamma)\) the area of the domain enclosed by \(\gamma\). The author proves the following: {\parindent=0.6 cm \begin{itemize}\item[(i)] if \(\gamma_{\alpha}\) is a convex curve, then \(A(\gamma_{\alpha}) \leq \cos^2(\alpha) A(\gamma)\), with equality if and only if \(\gamma\) is a Euclidean circle; \item[(ii)] for \(\alpha \in (0,{\pi \over 2})\), the curve \(\gamma\) is a circle if and only if \(\gamma_{\alpha}\) is a circle. \end{itemize}}
reviewed by: Gabriel Teodor Pripoae
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 6351390
zbMATH Keywords: planar convex curve
zbMATH Keywords: planar closed curve
zbMATH Keywords: evolutoid
zbMATH Keywords: secantoptic
MaRDI profile type: Publication
full work available at URL: s00010-013-0213-y
OpenAlex ID: W1998033772
cites work: Tanvolutes: Generalized Involutes
cites work: Q5690469
cites work: Über die Evolutoiden und Zwischenevolutoiden von Raumkurven
cites work: Some properties of secantoptics of ovals
cites work: A note on secantoptics
cites work: Q5338702
DOI: 10.1007/S00010-013-0213-Y
description: scientific article; zbMATH DE number 6351390