Open AccessRectifying curves and geodesics on a cone in the Euclidean 3-space
Author(s) -
BangYen Chen
Publication year - 2017
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.48.2017.2382
Subject(s) - geodesic , mathematics , cone (formal languages) , euclidean geometry , euclidean space , space (punctuation) , planar , geometry , plane (geometry) , mathematical analysis , position (finance) , pure mathematics , combinatorics , computer science , algorithm , computer graphics (images) , finance , economics , operating system
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.