Rectifying curves in the three-dimensional sphere

Lucas, Pascual; Ortega-Yagües, José Antonio

doi:10.1016/j.jmaa.2014.07.079

Cited by 27 publications

(39 citation statements)

References 5 publications

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“…7, we show that a rectifying curve γ = exp p (ρV ) in H 3 (−r) is characterized by the property that a certain function (depending on its speed v, κ and ρ) takes its minimum value, equal to k 2 V , among the hyperbolic curves with the same spherical projection V (k V denotes the geodesic curvature of V ). A similar result, which does not appear in [13], is stated for rectifying curves in the three-dimensional sphere S 3 (r).…”

Section: Introductionsupporting

confidence: 73%

“…In the case of rectifying curves in the three-dimensional sphere S 3 (r), see [13], we can obtain a similar result. The proof is analogous to the one of Theorem 8, and it is left to the reader.…”

Section: Mjomsupporting

confidence: 62%

“…Similar to [13,Theorem 7], we deduce that the nontrivial solutions of the above ODE are given by ρ(t) = r arg tanh(a sec(t + t 0 )). Hence, we conclude that γ(t) = exp p (ρ(t)V (t)) is a rectifying curve if and only if ρ(t) = r arg tanh(a sec(t + t 0 )), for some constants a and t 0 , with 0 < a 2 < 1.…”

supporting

confidence: 67%

“…(1) is that the straight line that connects x(s) with the point p is orthogonal to the principal normal line [i.e., the line starting at x(s) in the direction of N (s)]. This idea was used by the authors to define rectifying curves in the threedimensional sphere S 3 (r) [13]. In this paper, we study rectifying curves in the three-dimensional hyperbolic space H 3 (−r), and obtain several results of characterization and classification for that family of curves.…”

Section: Introductionmentioning

confidence: 99%

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Rectifying Curves in the Three-Dimensional Hyperbolic Space

Lucas

Ortega-Yagües

2015

Mediterr. J. Math.

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B. Y. Chen introduced rectifying curves in R 3 as space curves whose position vector always lies in its rectifying plane. Recently, the authors have extended this definition (as well as several results about rectifying curves) to curves in the three-dimensional sphere. In this paper, we study rectifying curves in the three-dimensional hyperbolic space, and obtain some results of characterization and classification for such kind of curves. Our results give interesting and significant differences between hyperbolic, spherical and Euclidean geometries.Mathematics Subject Classification. 53A04, 53A05.

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Section: Introductionsupporting

confidence: 73%

Section: Mjomsupporting

confidence: 62%

supporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

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Rectifying Curves in the Three-Dimensional Hyperbolic Space

Lucas

Ortega-Yagües

2015

Mediterr. J. Math.

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“…In view of these basic facts, the author asked the following very simple natural geometric question in [3] (see also [6] We simply call such a curve a rectifying curve in [3]. It is known that rectifying curves have many interesting properties (see, for instance, [3,4,5,6,7]). …”

Section: Rectifying Curvesmentioning

confidence: 99%

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

Chen¹

2017

Tamkang J. Math.

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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.

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

Silva

2018

Mediterr. J. Math.

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The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.Mathematics Subject Classification (2010) 53A04 · 53A05 · 53B20 · 53C21

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Rectifying curves in the three-dimensional sphere

Cited by 27 publications

References 5 publications

Rectifying Curves in the Three-Dimensional Hyperbolic Space

Rectifying Curves in the Three-Dimensional Hyperbolic Space

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

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

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