Special curves and ruled surfaces
type: Item
instance of: scholarly article
title: Special curves and ruled surfaces
zbMATH Open document ID: 1035.53024
author: Nobuko Takeuchi
author: Shyuichi Izumiya
published in: Beiträge zur Algebra und Geometrie
publication date: 2003-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/223081
full work available at URL: 15.html
review text: Cylindrical helices and Bertrand curves are studied from a new point of view: they are considered as curves on a ruled surface. It is shown that a ruled surface is the rectifying developable of a curve \(\gamma\) if and only if \(\gamma\) is the geodesic of the ruled surface which is transversal to rulings and whose Gaussian curvature vanishes along \(\pi\). As a consequence of this theorem, a new characterization of cylindrical surfaces is obtained. Another essential theorem states that a ruled surface is the principal normal surface of a space curve \(\gamma\) if and only if \(\gamma\) is the asymptotic curve of the ruled surface and has vanishing mean curvature along \(\gamma\). Applying this result, consequences on Bertrand curves and an interesting characterization of helicoids are deduced.
Mathematics Subject Classification ID: 53A25
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 1894854
zbMATH Keywords: cylindrical helix
zbMATH Keywords: Bertrand curve
zbMATH Keywords: ruled surfaces
reviewed by: József Szilasi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1894854