A method of the determination of a developable ruled surface
type: Item
instance of: scholarly article
title: A method of the determination of a developable ruled surface
zbMATH Open document ID: 0967.53010
DOI: 10.1016/S0094-114X(98)00059-7
author: Ömer Köse
published in: Mechanism and Machine Theory
publication date: 1999-09-15T00:00:00Z
review text: Let \(\widehat{\mathbf x}(t)= {\mathbf x}(t)+ \varepsilon{\mathbf x}^* (t)\) be a curve on the unit dual sphere which corresponds to the ruled surface \({\mathbf m}(t,u)= {\mathbf p}(t)+ u{\mathbf x}(t)\) (Study's Übertragungsprinzip), where \(\varepsilon^2= \varepsilon^3= \cdots=0\), \(|{\mathbf x} |\equiv 1\), and \({\mathbf x}^*: ={\mathbf p}\times{\mathbf x}\). The author considers the dual angles \(\widehat\varphi= \varphi+ \varepsilon\varphi^*\) and \(\widehat\theta= \theta+ \varepsilon \theta^*\) in order to obtain a parametric representation of the curve \(\widehat{\mathbf x}(t)\) and shows that the base curve is given by \({\mathbf p}(t)= ((P_3-\theta^*) \cos\theta \text{cotan} \varphi+ \varphi^* \sin\theta\), \((P_3- \theta^*) \sin\theta \text{ cotan }\varphi-\varphi^* \cos\theta, P_3)\). Taking \(P_3=\theta^*\) one has \({\mathbf p}(t)= (\varphi^* \sin\theta, -\varphi^*\cos \theta,\theta^*)\). For a developable ruled surface the linear differential equation \({dy\over dt}+ Ay+B=0\) is satisfied for \(y(t):= \tan\varphi\) and some coefficients \(A=A(t)\) and \(B=B(t)\). Using the above, the author shows that developable ruled surfaces can be found so that the base curve is a given curve \({\mathbf p}(t)\). He finally considers the example in the case that the given curve is the cylindrical helix.
Mathematics Subject Classification ID: 53A25
Mathematics Subject Classification ID: 53A40
zbMATH DE Number: 1331551
zbMATH Keywords: Study's Übertragungsprinzip
zbMATH Keywords: ruled surface
reviewed by: Stylianos Stamatakis
MaRDI profile type: Publication
Wikidata QID: Q126589563
description: scientific article; zbMATH DE number 1331551