Characterization of the slant helix as successor curve of the general helix
type: Item
instance of: scholarly article
zbMATH Open document ID: 1316.53004
author: Anton Menninger
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1411.0550
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 6366155
zbMATH Keywords: general helix
zbMATH Keywords: slant helix
zbMATH Keywords: curves of constant precession
zbMATH Keywords: Frenet equations
zbMATH Keywords: Frenet curves
zbMATH Keywords: Frenet frame
zbMATH Keywords: Bishop frame
zbMATH Keywords: natural equations
zbMATH Keywords: successor curves
zbMATH Keywords: Salkowski curves
zbMATH Keywords: closed curves
MaRDI profile type: Publication
arXiv ID: 1411.0550
arXiv classification: math.DG
title: Characterization of the slant helix as successor curve of the general helix
published in: International Electronic Journal of Geometry
review text: Given a pair of unit speed curves \((\gamma,\delta)\) in Euclidean \(3\)-space \(\mathbb{E}^3\), the author calls \(\delta\) a \textit{successor curve} for \(\gamma\) if the normal vector field of \(\delta\) equals the tangent vector field of \(\gamma\). Recall that a curve is called a \textit{general helix} if its tangent vector field spans a constant angle with a fixed line in \(\mathbb{E}^3\). Likewise, a curve is called a \textit{slant helix} if its normal tangent vector field spans a constant angle with a fixed line in \(\mathbb{E}^3\).NEWLINENEWLINEIn the paper under review the author gives a description of the family of successor curves of a given curve \(\gamma\) in terms of its Frenet frame and its curvature and torsion. Furthermore, he shows that a regular \(C^2\) space curve is a general helix if and only if it is a successor curve of a plane curve (and not itself a plane curve). He also provides a characterisation of slant helices.
reviewed by: Thomas Mettler
description: scientific article; zbMATH DE number 6366155