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

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.

Keywords


Geodesic; cone; circular cone; rectifying curve

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References


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DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.2382

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