Evolutes of hyperbolic plane curves
type: Item
instance of: scholarly article
title: Evolutes of hyperbolic plane curves
zbMATH Open document ID: 1065.53022
author: Donghe Pei
author: Takashi Sano
author: Erika Torii
author: Shyuichi Izumiya
publication date: 2004-10-01T00:00:00Z
review text: Let \(\mathbb{R}^3_1\) be the Minkowski 3-space, or pseudo-Euclidean space, i.e. the 3-dimensional vector-space \(\mathbb{R}^3\) equipped with the scalar product \(\langle x,y\rangle=-x_1y_1+x_2y_2+x_3y_3\). The Lorentzian sphere: \(\langle x,x\rangle=-1\), a two-sheet-hyperboloid, contains as one of its sheets the surface \(H^2_+\): \(\langle x,x\rangle=-1\), \(x_1\geq 1\) which is adopted as the model of the hyperbolic plane and (wrongly) called ``hyperbola''. The authors develop the Frenet-Serret-type formula for hyperbolic plane curves and study in particular the following properties of these curves: hyperbolic invariants, osculating pseudo-circles, the hyperbolic evolute and its singularities. The case when a point of the hyperbolic evolute is an ordinary cusp is characterized (theorem 5.3). The last section describes how one can draw the picture of the hyperbolic evolute of a curve on the Poincaré disk: fig. 1 shows a family of Euclidean ellipses and their hyperbolic evolutes. The authors mention -- what can be seen clearly in fig. 1 -- that the four cusps theorem does not hold in \(H^2_+\) (although the four vertex theorem does).
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 2104913
zbMATH Keywords: singularity
zbMATH Keywords: Lorentz group
zbMATH Keywords: Poincaré disk
MaRDI profile type: Publication
full work available at URL: s10114-004-0301-y
OpenAlex ID: W1964713215
cites work: Q4003405
DOI: 10.1007/S10114-004-0301-Y
published in: Acta Mathematica Sinica, English Series
Recommended article: A Hyperbolic Theory for the Evolution of Plane Curves
Recommended article: Evolutes of conics in the quasi-hyperbolic and the hyperbolic plane
Recommended article: On evolutoids of planar convex curves
Recommended article: Evolutoids and pedaloids of plane curves
Recommended article: FOCAL SURFACES AND EVOLUTES OF CURVES IN HYPERBOLIC SPACE
Recommended article: The Flow-geodesic Curvature and the Flow-evolute of Hyperbolic Plane Curves
Recommended article: Hyperbolic curve flows in the plane
description: scientific article; zbMATH DE number 2104913