Bertrand Curves Research Articles

SOME ISOTROPIC CURVES AND REPRESENTATION IN COMPLEX SPACE ℂ3

In this paper, we give a representation formula for an isotropic curve with pseudo arc length parameter and define the structure function of such curves. Using the representation formula and the Frenet formula, the isotropic Bertrand curve and k-type isotropic helices are characterized in the 3-dimensional complex space <TEX>$\mathbb{C}^3$</TEX>.

BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE

In this paper, we generalize some results about Bertrand curves and Razzaboni surfaces in Euclidean 3-space to the case that the ambient space is Minkowski 3-space. Our discussion is divided into three different cases, i.e., the parent Bertrand curve being timelike, spacelike with timelike principal normal, and spacelike with spacelike principal nor- mal. For each case, first we show that Razzaboni surfaces and their mates are related by a reciprocal transformation; then we give Backlund trans- formations for Bertrand curves and for Razzaboni surfaces; finally we prove that the reciprocal and Backlund transformations on Razzaboni surfaces commute.

On the Differential Geometric Elements of the Involute $${\tilde{\rm D}}$$ D ~ -Scroll in E 3

Deriving curves based on the other curves is a subject in geometry. Involute-evolute curves, Bertrand curves are this kind of curves. By using the similiar method we produce a new ruled surface based on the other ruled surface. In [14], \({\tilde{D}}\)-scroll, which is known as the rectifying developable surface, of any curve \({\alpha}\) and the involute\({\tilde{D}}\)-scroll of the curve \({\alpha}\) are already defined, \({\mathbb{E}^{3}}\). In this paper, we consider these special ruled surfaces \({\tilde{D}}\)-scroll and involute\({\tilde{D}}\)-scroll, associated to a space curve \({\alpha}\) with curvature \({k_{1} \neq 0}\) and involute \({\beta}\). We will examine the differential geometric elements (such as , Weingarten map S, curvatures K and H) of the involute\({\tilde{D}}\)-scroll and \({\tilde{D}}\)-scroll relative to each other. Further we will examined the fundamental forms too.

On Space-Like Constant Slope Surfaces And Bertrand Curves In Minkowski 3-Space

Abstract In the present paper, we define the notions of Lorentzian Sabban frames and de Sitter evolutes of the unit speed space-like curves on de Sitter 2-space 𝕊21. In addi- tion, we investigate the invariants and geometric properties of these curves. Afterwards, we show that space-like Bertrand curves and time-like Bertrand curves can be constructed from unit speed space-like curves on de Sitter 2-space 𝕊21 and hyperbolic space ℍ2, respec- tively. We obtain the relations between Bertrand curves and helices. Also we show that pseudo-spherical Darboux images of Bertrand curves are equal to pseudo-spherical evo- lutes in Minkowski 3-space ℝ31. Moreover we investigate the relations between Bertrand curves and space-like constant slope surfaces in ℝ31. Finally, we give some examples to illustrate our main results.

New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

On Pseudo Null Bertrand Curves in Minkowski Space-time

In this paper, we prove that there are no pseudo null Bertrand curve with curvature functions k1(s) = 1; k2(s) 0 and k3(s) other than itself in Minkowski space- time E 4 and by using the similar idea of Matsuda and Yorozu (13), we define a new kind of Bertrand curve and called it pseudo null (1; 3)-Bertrand curve. Also we give some characterizations and an example of pseudo null (1; 3)-Bertrand curves in Minkowski space- time.

Generalized Null Bertrand Curves In Minkowski Space-Time

Abstract Çöken and ÇIFTCI proved that a null Cartan curve in Minkowski space-time E41 is a null Bertrand curve if and only if k2 is nonzero constant and k3 is zero. That is, the null curve with non-zero curvature k2 is not a Bertrand curve in Minkowski space-time E41. So, in this paper we defined a new type of Bertrand curve in Minkowski space-time E41 for a null curve with non-zero curvature k3 by using the similar idea of generalized Bertrand curve given by Matsuda and Yorozu and we called it a null (1, 3)-Bertrand curve. Also, we proved that if a null curve with non-zero curvatures in Minkowski space-time E41 is a null (1, 3)-Bertrand curve then it is a null helix. We give an example of such curves.

Time-Like Constant Slope Surfaces and Space-Like Bertrand Curves in Minkowski 3-Space

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space \( {\mathbb{S}}_{1}^{2} \) and introducing space-like height function on the unit speed time-like curves on \( {\mathbb{S}}_{1}^{2} , \) the invariants of the unit speed time-like curves on \( {\mathbb{S}}_{1}^{2} , \) and geometric properties of de Sitter evolutes of the unit speed time-like curves on \( {\mathbb{S}}_{1}^{2} \) are studied. A relation between space-like Bertrand curves and helices is obtained. De Sitter Darboux images of space-like Bertrand curves are equal to de Sitter evolutes. The relations between time-like constant slope surfaces lying in the space-like cone and space-like Bertrand curves in Minkowski 3-space \( {\mathbb{R}}_{1}^{3} \) are obtained.

CURVE COUPLES AND SPACELIKE FRENET PLANES IN MINKOWSKI 3-SPACE

Abstract. In this study, we have investigated the possibility ofwhether any spacelike Frenet plane of a given space curve in Minkowski3-space E 31 also is any spacelike Frenet plane of another space curvein the same space. We have obtained some characterizations of agiven space curve by considering nine possible case. 1. IntroductionIn the theory of curves in Euclidean space, one of the importantand interesting problem is characterization of a regular curve. In thesolution of the problem, the curvature functions k 1 (or {) and k 2 (or ˝)of a regular curve have an e ective role. For example: if k 1 = 0 = k 2 ,then the curve is a geodesic or if k 1 = constant 6= 0 and k 2 = 0;thenthe curve is a circle with radius (1=k 1 ), etc. Thus we can determine theshape and size of a regular curve by using its curvatures.Another way in the solution of the problem is the relationship betweenthe Frenet vectors of the curves (see [6]). For instance Bertrand curves:In 1845, Saint Venant (see [6] and [8]) proposed the question whetherupon the surface generated by the principal normal of a curve, a secondcurve can exist which has for its principal normal the principal normalof the given curve. This question was answered by Bertrand in 1850 ina paper which he showed that a necessary and sucient condition forthe existence of such a second curve is that a linear relationship withconstant coecients shall exist between the rst and second curvaturesof the given original curve. In other word, if we denote rst and second

English

Starting from ideas and results given by &Ouml;zkaldi, İlarslan and Yaylı in (2009), in this paper we investigate Bertrand curves in three dimensional dual space D3. We obtain the necessary characterizations of these curves in dual space D3. As a result, we find that the distance between two Bertrand curves and the dual angle between their tangent vectors are constant. Also, well known characteristic property of Bertrand curve in Euclid space E3&nbsp;which is the linear relation between its curvature and torsion is satisfied in dual space as&nbsp;&lambda;.&kappa;(S) + &mu;.&tau;(S) = 1.&nbsp;We show that involute curves, which are the curves whose tangent vectors are perpendicular, of a curve constitute Bertrand pair curves. Key words: Bertrand curves, involute-evolute curves, dual space. &nbsp; 2000 Mathematics Subject Classification: 53A04 , 53A25 , 53A40.

Curves in Three Dimensional Riemannian Space Forms

In three dimensional Riemannian space forms, introducing a natural moving frame, we define the associate curve of a curve and study the properties and relations of a curve and its associate curve. We state necessary and sufficient condition that a Frenet curve is a Bertrand curve in three dimensional Riemannian space forms, especially in a Riemannian 3-dimensional sphere and in a 3-dimensional hyperbolic space, resp. At the same time we give an explicit expression of the partner curve of a Bertrand curve.

Generalized spacelike Bertrand curves in Minkowski 5-space

In this paper, we give the definition of the generalization of spacelike Bertrand curves and obtain the characterization of these curves in Minkowski 5-space . Moreover, we define three types special spacelike Bertrand curves in Minkowski 5-space

Bertrand Curves of AW(k)-Type in the Equiform Geometry of the Galilean Space

We consider curves of AW(k)-type (1≤k≤3) in the equiform geometry of the Galilean spaceG3. We give curvature conditions of curves of AW(k)-type. Furthermore, we investigate Bertrand curves in the equiform geometry ofG3. We have shown that Bertrand curve in the equiform geometry ofG3is a circular helix. Besides, considering AW(k)-type curves, we show that there are Bertrand curves of weak AW(2)-type and AW(3)-type. But, there are no such Bertrand curves of weak AW(3)-type and AW(2)-type.

Special Bertrand Curves in 4D Galilean Space

The generalization of Bertrand curves in Galilean 4-space is introduced and the characterization of the generalized Bertrand curves is obtained. Furthermore, it is proved that no special curve is a classical Bertrand curve in Galilean 4-space such that the notion of classical Bertrand curve is definite only in three-dimensional spaces.

CURVES ON THE UNIT 3-SPHERE S3(1) IN EUCLIDEAN 4-SPACE ℝ4

We show many examples of curves on the unit 2-sphere <TEX>$S^2(1)$</TEX> in <TEX>$\mathbb{R}^3$</TEX> and the unit 3-sphere <TEX>$S^3(1)$</TEX> in <TEX>$\mathbb{R}^4$</TEX>. We study whether its curves are Bertrand curves or spherical Bertrand curves and provide some examples illustrating the resultant curves.

BERTRAND CURVES IN NON-FLAT 3-DIMENSIONAL (RIEMANNIAN OR LORENTZIAN) SPACE FORMS

Let M 3(c) denote the 3-dimensional space form of index q = 0,1, and constant curvature c 6 0. A curveimmersed in M 3(c) is said to be a Bertrand curve if there exists another curveand a one-to-one correspondence betweenandsuch that both curves have common principal normal geodesics at corresponding points. We obtain charac- terizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non- null Bertrand curves in M3(c) correspond with curves for which there exist two constants � 6 0 and µ such that �� + µ� = 1, whereand � stand for the curvature and torsion of the curve. As a consequence, non-null helices in M 3(c) are the only twisted curves in M 3(c) having in- finite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

On Helices and Bertrand Curves in Euclidean 3-Space

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its the tangent indicatrix and binormal indicatrix are both Bertrand curves and circular helices. Similarly, in case of a space curve is a slant helix, we demonstrate that the curve corresponding to the spherical image of its the principal normal indicatrix is both a Bertrand curve and a circular helix.

A variational characterization and geometric integration for Bertrand curves

In this paper, we introduce a class of functionals, in the three-dimensional semi-Euclidean space \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3, having an energy density that depends only on curvature and whose moduli space of trajectories consists of LW-curves, i.e., curves with curvature κ and torsion τ for which there are three real constants λ, μ, ρ such that λκ + μτ = ρ, with λ2 + μ2 &amp;gt; 0. This family of curves includes plane curves, helices, curves of constant curvature, curves of constant torsion, Lancret curves (also called generalized helices), and Bertrand curves. We present an algorithm to construct Bertrand curves in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3 by using an arclength parametrized curve in a totally umbilical surface \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2$\end{document}S2, \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^2_1$\end{document}S12, or \documentclass[12pt]{minimal}\begin{document}$\mathbb {H}^2$\end{document}H2 and prove that every Bertrand curve in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3_q$\end{document}Rq3 can be obtained in this way. A second algorithm is presented for the construction of LW-curves by using a curve of constant slope in the ruled surface Sα whose directrix is a certain curve α with non-zero curvature and whose rulings are generated by its modified Darboux vector field.

Spherıcal ınvolutes of the fıxed pole curve on the tımelıke bertrand curve couple

In this paper, it has been showed that every single indicatrix of tangents and indicatrix of binormals of the curve, (alpha^*) are spherical involutes of the fixed pole curve, (C^*) by finding a transition link of the timelike Bertrand curve couple through Frenet frames.

Characterizations of Special Curves

In this study, the new characterizations of special curves are investigated without using the curvatures of these special curves: general helices, slant helices, Bertrand curves, Mannheim curves. The curvatures are given by the help of the norms of the derivatives of Frenet vectors.