A new approach to Bertrand curves in Euclidean 3-space
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Journal of Geometry
A new approach to Bertrand curves in Euclidean 3-space ˙ C ¸ etin Camci, Ali U¸cum, and Kazım Ilarslan Abstract. In this article, a new approach is given for Bertrand curves in 3-dimensional Euclidean space. According to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Bertrand curve in E3 . In addition, the related examples and graphs are given by showing that general helices and antiSalkowski curves can be Bertrand curves or their mates, which is their new characterization. Mathematics Subject Classification. 53A04. Keywords. Bertrand curves, General helices, Anti-Salkowski curves, Euclidean 3-space.
1. Introduction In the Euclidean 3-space E3 , a curve is called a general helix if its tangent vector makes a constant angle with a fixed straight line (the axis of the general helix). The well-known result stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845 (for details see [4,5]) puts forth that a necessary and sufficient condition that a curve be a general helix is that the ratio of curvature to torsion be constant. Also it is known that a curve is called a circular helix if both curvatures k1 and k2 are non-zero constant. Circular helices geometrically appear as geodesic in right cylinders shaped on circle. The geodesics of a right cylinder, with arbitrary cross section, are called general or Lancret helices. In addition, Izumiya and Takeuchi have introduced the concept of slant helix by saying that the normal lines make a constant angle with a fixed straight line. They characterize a slant helix if and only if the geodesic curvature κ2 κ21 κg = κ1 (κ2 + κ2 )3/2 1 2 0123456789().: V,-vol
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of the principal image of the principal normal indicatrix is a constant function [3]. A family of curves with constant curvature but non-constant torsion is called Salkowski curves and a family of curves with constant torsion but non-constant curvature is called anti-Salkowski curves [1,8]. In [6], Monterde studied some of characterizations of these curves and he proved that the principal normal vector makes a constant angle with fixed straight line. So that: Salkowski and anti-Salkowski curves are important examples of slant helices. On the other hand, in 1845, Saint Venant [7] put forward the question whether the principal normal of a curve is the principal normal of another’s on the surface generated by the principal normal of the given one. Bertrand gave an answer to this question in the paper published in 1850 [2]. He proved that a necessary and sufficient condition for the existence of such a second curve is required in fact a linear relationship calculated with constant coefficients should exist between the first and second curvatures of the given original curve. In other words, we have λκ1 + μκ2 = 1, λ, μ ∈ R where κ1 and κ2 is denoted by the first and second curvatures of a given curve respectively. Since 1850, after the paper of Bertrand, the pairs of curves lik
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