Ruled Surfaces and Tangent Bundle of Unit 2-Sphere of Natural Lift Curves

Abstract

Highlights• A unique ruled surface is corresponded to the natural lift curve.• Properties of ruled surfaces generated by natural lift curves are examined. • A method is given for modelling motions on ̅ instead of 2 .

Singularities of the Ruled Surfaces According to RM Frame and Natural Lift Curves

“…is the ruled surface in 3  corresponding to the dual curve     2 Γ(s) = q(s) + εv(s) DS (or to the natural lift curve  Γ(s) TM ) in [19]. Considering the ruled surface given in Eq.…”

“…In the light of this study, a one-to-one correspondence between and is mentioned in [18]. In that study, according to E. Study mapping, to each curve on corresponds a ruled surface in Euclidean 3-space, Furthermore, exploiting this relation, each curve on corresponds a ruled surface in Then, inspired by [18], the isomorphism among , the subset of the tangent bundle of unit 2-sphere, and the ruled surface generated by natural lift curves in are examined in [19]. Furthermore, the developability condition of this ruled surface is given in the same study.…”

“…Definition 1 For the curve γ , γ is called the natural lift of γ on TM, which produces in the following equation, [19]:…”

B-Lift curves and its ruled surfaces

“…A ruled surface is defined by a straight line that is moving along a curve [1]. Many mathematicians have studied the ruled surfaces [2][3][4][5][6][7][8]. E. Ergün and M. C ¸alışkan [2] created ruled surfaces by accepting the natural lift of a curve as the base curve and they characterized these surfaces.…”

Bertrand Curves and B-Lift Curves in Lorentzian 3-Space

“…The natural lift curve is defined as the curve drawn by the endpoints of the unit tangent vector at each point of a given curve. Many mathematicians have studied on natural lift curves [2][3][4][5][6]. The Frenet vectors of the natural lift curve were introduced by C ¸alıs ¸kan and Ergün with regard to the Frenet vectors of the main curve [5].…”