Generalized Helices in an Ordinary Vn
Published online by Cambridge University Press: 24 October 2008
Extract
A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 1 , January 1941 , pp. 14 - 28
- Copyright
- Copyright © Cambridge Philosophical Society 1941
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