RECTIFYING CURVES AND GEODESICS ON A CONE IN THE EUCLIDEAN 3-SPACE

BANG-YEN CHEN

《Tamkang Journal of Mathematics》 48卷2期 (2017/06) Pp. 209-214

https://doi.org/10.5556/j.tkjm.48.2017.2382

摘要

A twisted curve in the Euclidean 3-space 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 E^3, not necessary a circular one, via rectifying curves. Our main result states that a curve on a cone in 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 E^3 are portions of rulings.

關鍵字

Geodesic ； cone ； circular cone ； rectifying curve