Rectifying curves and geodesics on a cone in the Euclidean 3-space

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Published: Jun 30, 2017
DOI: https://doi.org/10.5556/j.tkjm.48.2017.2382
Keywords:
Geodesic cone circular cone rectifying curve

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Bang-Yen Chen

Abstract

A twisted curve in the Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lie in its rectifying plane. In this article we study geodesics on an arbitrary cone in $\mathbb E^3$, not necessary a circular one, via rectifying curves. Our main result states that a curve on a cone in $\mathbb E^3$ is a geodesic if and only if it is either a rectifying curve or an open portion of a ruling. As an application we show that the only planar geodesics in a cone in $\mathbb E^3$ are portions of rulings.

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How to Cite
Chen, B.-Y. (2017). Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang Journal of Mathematics, 48(2), 209–214. https://doi.org/10.5556/j.tkjm.48.2017.2382
Issue
Vol. 48 No. 2 (2017)
Section
Papers
Author Biography

Bang-Yen Chen

Department of Mathematics,Michigan State University, 619 Red Cedar Road, East Lansing,Michigan 48824–1027,U.S.A.

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