The Frenet Vector Fields and the Curvatures of the Natural Lift Curve

Ergün, Evren; Bi̇li̇ci̇, Mustafa; Çalışkan, Mustafa

doi:10.18052/www.scipress.com/bsmass.2.38

Cited by 13 publications

(23 citation statements)

References 5 publications

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“…The relation among 2 DS , 2 TS and non-cylindirical ruled surfaces was given, see [12]. In the light of this study, the relationship between tangent bundle of unit 2-sphere, 2 , TS and unit dual sphere, 2 DS was given [13]. In that study, the correspondence between an each curve on 2 TS and ruled surface in 3 was given.…”

Section: Introductionmentioning

confidence: 85%

“…Thorpe's book [1]. In 3 , the properties of natural lift curve of a given curve were studied [2]. Frenet vector fields, the first, and second curvatures of the natural lift curve of a given curve were calculated by using the angle between Darboux vector field and the binormal vector field of the given curve [3].…”

Section: Introductionmentioning

confidence: 99%

“…Ruled surfaces have been extensively investigated in geometry, surface design, computer-aided geometric design, physics, etc. In 3 , a ruled surface is a curved surface represented by moving a straight line along a space curve called the base curve [4]. In literature, ruled surfaces have been widely studied by different authors [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…Dual numbers were introduced by W. K. Clifford in 1873. Then E. Study constructed the correspondence between the geometry of lines in 3 and unit dual sphere, 2 DS by defining a mapping called E. Study mapping. This mapping offers that there exists a one-toone correspondence between the oriented straight lines in 3 and the points on 2 DS , see [11].…”

Section: Introductionmentioning

confidence: 99%

“…However, there is limited research into ruled surfaces generated by the natural lift curves in 3 1 with the isomoprhisms between the subsets of pseudo-sphere and Lorentzian unit sphere, 2 1 or hyperbolic unit sphere, 2 . The remainder of this paper is divided into four sections: Section 2 presents basic definitions and theorems about the topic.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Ruled Surfaces and Tangent Bundle of Pseudo-Sphere of Natural Lift Curves

Karaca

Çalışkan

2020

J. Sci. Arts

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In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, TM, and Lorentzian unit sphere, S12 is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, TM, and hyperbolic unit sphere, H2 is given. According to E. Study mapping, any curve on S12 or H2 corresponds to a ruled surface in R13. By constructing these isomorphisms, we correspond to any natural lift curve on TM or TM a unique ruled surface in R13. Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.

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Section: Introductionmentioning

confidence: 85%