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Characterizations of slant helices in Euclidean 3-space

In this paper we investigate the relations between a general helix and a slant helix. Moreover, we obtain some differential equations which they are characterizations for a space curve to be a slant helix. Also, we obtain the slant helix equations and its Frenet aparatus. | Turk J Math 34 (2010) , 261 – 273. ¨ ITAK ˙ c TUB doi:10.3906/mat-0809-17 Characterizations of slant helices in Euclidean 3-space ˙ L. Kula, N. Ekmekci, Y. Yaylı and K. Ilarslan Abstract In this paper we investigate the relations between a general helix and a slant helix. Moreover, we obtain some differential equations which they are characterizations for a space curve to be a slant helix. Also, we obtain the slant helix equations and its Frenet aparatus. Key Words: Slant helix, genaral helix, spherical helix, tangent indicatrix, principal normal indicatrix and binormal indicatrix. 1. Introduction In differential geometry, a curve of constant slope or general helix in Euclidean 3-space R3 is defined by the property that the tangent makes a constant angle with a fixed straight line (the axis of the general helix). A classical result stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845 (see [11, 13] for details) is: A necessary and sufficient condition that a curve be a general helix is that the ratio of curvature to torsion be constant. If both of κ and τ are non-zero constant it is, of course, a general helix. We call it a circular helix. Its known that straight line and circle are degenerate-helix examples (κ = 0 , if the curve is straight line and τ = 0 , if the curve is a circle). The study of these curves in R3 as spherical curves is given by Monterde in [12] . The Lancret theorem was revisited and solved by Barros (in [2] ) in 3-dimensional real space forms by using killing vector fields as along curves. Also in the same space-forms, a characterization of helices and Cornu spirals is given by Arroyo, Barros and Garay in [1] . On the studies of general helices in Lorentzian space forms, Lorentz-Minkowski spaces, semi-Riemannian manifolds, we refer to the papers [3, 4, 5, 6, 7, 9] . In [8] , A slant helix in Euclidean space R3 was defined by the property that the principal normal makes a constant angle with a fixed direction. Moreover, .