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

Abstract

Abstract.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.

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

“…This makes sense due to the double nature of R m+1 as both a manifold and as a tangent space. In fact, this problem has to do with the more general quest of studying curves that lie on a given (moving) plane generated by two chosen vectors of a moving trihedron, e.g., one would define osculating, normal or rectifying curves as those curves whose position vector, up to a translation, lies on their osculating, normal or rectifying planes, respectively [8,10]: osculating curves are the plane curves (if we substitute the principal normal by an RM vector field, we still have a characterization for plane curves [11]) and rectifying curves are precisely geodesics on a cone [9,10]. This equivalence is no longer valid in other geometries.…”

On rectifying curves in Euclidean 3-space

“…Therefore, not all curves that are dilations of unit speed curve y(t) on S 2 of the type α(t) = a sec(t + t 0 )y(t) are rectifying curves. This suggests finding the Frenet-Serret apparatus for the curve α(t) = a sec(t + t 0 )y(t) , which will help us to exclude those unit speed curves on S 2 , which do not allow α to be a rectifying curve (also pointed out independently in [3]).…”

“…The condition aκ − bτ = c with a 2 + b 2 ̸ = 0 and c ̸ = 0 given in Theorem 4.1(a) has been used by Lucas and Ortega-Yagües in[9] for their study of Betrand curves in the Euclidean 3-space E3 or Lorentz-Minkowski 3-space L Remark Let α(t) be the unit speed curve of Theorem 4.1 with κ ′ ̸ = 0 such that the centrode d(t) of α is a rectifying curve. Then asĉκ ′ = τ ′ , we getĉκ = τ + c 1 for a constant c 1 ̸ = 0 (as τ /κ is nonconstant), that is,ĉκ − τ = c 1 .…”

Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

“…Moreover, he completely classified in [20] rectifying curves in E 3 . Furthermore, he proved in [21] that a curve on a general cone (not necessarily a circular one) in E 3 is a geodesic if and only if it is a rectifying curve or an open portion of a ruling of the cone. In [22], several interesting links between rectifying curves, centrodes and extremal curves were established by B.-Y.…”