Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

Etayo, Fernando

doi:10.3906/mat-1609-86

Cited by 8 publications

(19 citation statements)

References 22 publications

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“…The basic idea here is that n i rotates only the necessary amount to remain normal to t: in fact, n i is parallel transported along α with respect to the normal connection [14]. Due to their minimal twist, RM frames are of importance in applications, such as in computer graphics and visualization [16,33], sweep surface modeling [3,27,29], and in differential geometry as well [2,11,12,15], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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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“…where . An RM frame, a parallel frame, a natural frame, a Bishop frame or an adapted frame is a moving orthonormal frame {T(s), (s)}, i = 1,2,...,n-1 along , where T(s) is the tangent vector to at the point ( ) and (s) = { 1 (s), 2 (s),..., −1 (s)} are RM vector fields, see [2,3]. If = ( ), '(s) = T and u(s) is an RM vector field, then ∧ is an RM vector field.…”

Section: Preliminariesmentioning

confidence: 99%

“…Any orthonormal basis { ′ ( 0 ), 1 ( 0 ), … , −1 ( 0 ) } at a point ( 0 ) expresses a unique RMF along the curve . Hence, such an RMF is uniquely designated modula a rotation in −1 [1,2,3,4,5]. Recently, RM frames is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, etc.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, all the results of RM vectors and frames are generalized to curves immersed in Riemannian manifolds. RM vector fields along a curve immersed into a Riemannian manifold are studied when the ambient manifold is the Euclidean 3-space, the Hyperbolic 3-space and a Kähler manifold [3]. But, we have seen that an RMF has not been studied much in the papers.…”

Section: Introductionmentioning

confidence: 99%

“…2 and 3 are the principal curvatures of . Such that 1 ( ) = −〈 ′( ), 〉, 2 ( ) = −〈 3 ′( ), 〉 and 3 ( ) = −〈 4 ′( ), 3 〉, see[9].Definition A normal vector field V = V(s) over a curve = ( ) in is said to be relatively parallel or Rotation minimizing (RM) if the derivative V'(s) is proportional to ′( ), see[2,3].Remark Let a normal vector field V = V(s) over a curve = ( ) in 3 . Then, the following statements can be written.…”

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confidence: 99%

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Rotation Minimizing Frame and Rectifying Curves in E_1^n

Keskin¹,

Yaylı²

2021

JAMC

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In this paper, some applications of a Rotation minimizing frame (RMF) are studied in E 1 4 and in E 1 n for timelike, spacelike curves. Firstly, in E 1 4 , a Rotation minimizing frame (RMF) is obtained on the timelike and spacelike direction curves ∫ N(s) ds. The features of this Rotation minimizing frame are expressed. Secondly, it is determined when the timelike and spacelike curves can be rectifying curves. In addition, it has been investigated the conditions under which timelike and spacelike curves can be sphere calcurves. Then, a new characterization of rectifying curves is given, similar to the characterization of spherical curves. Finally, this Rotation minimizing frame is generalized in E 1 n for timelike, spacelike curves. In E 1 n , the conditions being a spherical curve and arectifying curve are given thank to this frame for timelike and spacelike curves. Also, a relationship between the spherical curve and the rectifying curve is stated. It is shown that the coefficients used in obtaining rectifying curves are constant numbers.

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On calculation of the pitch and angle of pitch of a closed ruled surface of dimension (k + 1) using a rotation minimizing frame in Euclidean space ℝn

Keskin

Altın

Ekmekci

et al. 2021

Math Methods in App Sciences

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In this paper, first, the pitch and the angle of pitch of a closed ruled surface of dimension (k + 1) are calculated according to a rotation minimizing frame in Euclidean space R 4 . To calculate this pitch and this angle of pitch of the closed ruled surface of dimension (k + 1), three different ruled surfaces are taken according to the rotation minimizing frame. Therefore, for these three different ruled surfaces, three different situations are found. Second, by taking any curve in four-dimensional Euclidean space, the closed ruled surface of this curve, its orthogonal trajectory, the pitch, and the angle of pitch of this closed ruled surface are obtained according to the rotation minimizing frame. Finally, the pitch and the angle of pitch of the closed ruled surface of dimension (k + 1) are generalized to n-dimensional Euclidean space with the help of the rotation minimizing frame.

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Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

Cited by 8 publications

References 22 publications

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

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

Rotation Minimizing Frame and Rectifying Curves in E_1^n

On calculation of the pitch and angle of pitch of a closed ruled surface of dimension (k + 1) using a rotation minimizing frame in Euclidean space ℝn

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