Construction of Curve Pairs and Their Applications

Önder, Mehmet

doi:10.1007/s40010-019-00643-2

Cited by 3 publications

(3 citation statements)

References 14 publications

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“…In [9], a method to create a Bertrand curve using a spherical curve with Sabban frame was introduced. Also, in [17], some methods to create special curves such as helix, slant helix, Bertrand curves, and Mannheim curves were introduced.…”

Section: Advances In Mathematical Physicsmentioning

confidence: 99%

Position Vectors of the Natural Mate and Conjugate of a Space Curve

Alghanemi

Khan

2023

Advances in Mathematical Physics

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The concept of the natural mate and the conjugate curves associated to a smooth curve in Euclidian 3-space were introduced initially by Dashmukh and others. In this paper, we give some extra results that add more properties of the natural mate and the conjugate curves associated with a smooth space curve in E 3 . The position vectors of the natural mate and the conjugate of a given smooth curve are investigated. Also, using the position vector of the natural mate, the necessary and sufficient condition for a smooth given curve to be a Bertrand curve is introduced. Moreover, a new characterization of a general helix is introduced.

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Section: Advances In Mathematical Physicsmentioning

confidence: 99%

Position Vectors of the Natural Mate and Conjugate of a Space Curve

Alghanemi

Khan

2023

Advances in Mathematical Physics

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“…Similarly, Macit and Düldül have defined W -direction curve and W -donor curve in E 3 , where W is unit Darboux vector of the reference curve [10]. Later, the author has defined Bertrand direction curves, Mannheim direction curves and involute-evolute direction curves in E 3 and introduced relations between those curves and some special curves such as helices and slant helices [14].…”

Section: Introductionmentioning

confidence: 99%

Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

Önder¹

2022

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this study, we define (1,3)-Bertrand-direction curve and (1,3)-Bertrand-donor curve in the 4-dimensional Euclidean space $E^{4}$. We introduce necessary and sufficient conditions for a special Frenet curve to have a (1,3)-Bertrand-direction curve. We introduce the relations between Frenet vectors and curvatures of these direction curves. Furthermore, we investigate whether (1,3)-evolute-donor curves in $E^{4}$ exist and show that there is no (1,3)-evolute-donor curve in $E^{4}$ .

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“…Well-known pairs of curves are Involute-Evolute, Parallel, Bertrand, Mannheim, and Natural mates. Characterizations of these curves mates with some properties can be found in various papers : [2,6,12,13,14,15,20,23,25,27,31,32,39,42]. Some of these curves mates have been generalized to larger dimensions and have been studied by many authors [11,17,19,28,35].…”

Section: Introductionmentioning

confidence: 99%

A New Generalization of Some Curve Pairs

ÇELİK

Özdemir²

2022

International Electronic Journal of Geometry

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In this study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $\gamma$ with the tangents of the curves. We examine the relations between torsions and curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=\cos (\theta/2)-\mathbf{i}\sin (\theta/2)\cos \gamma -\mathbf{j}\sin (\theta/2)\sin \gamma$, where $\theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.

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Construction of Curve Pairs and Their Applications

Cited by 3 publications

References 14 publications

Position Vectors of the Natural Mate and Conjugate of a Space Curve

Position Vectors of the Natural Mate and Conjugate of a Space Curve

Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

A New Generalization of Some Curve Pairs

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