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Characterizations of slant helices in Euclidean 3-space
(Scientific Technical Research Council Turkey-Tubitak, 2010) Kula, L.; Ekmekci, N.; Yayli, Y.; Ilarslan, K.
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.
ON THE EXPLICIT CHARACTERIZATION OF CURVES ON A (n - 1)-SPHERE IN Sn
(Int Electronic Journal Geometry, 2013) Camci, C.; Kula, L.; Ilarslan, K.; Hacisalihoglu, H. H.
In (n+1)-dimensional Euclidean space E-n+(1), harmonic curvatures and focal curvatures of a non-degenerate curve were defined by Ozdamar and Hacisalihoglu in [7] and by Uribe-Vargas in [9], respectively. In this paper, we investigate the relations between the harmonic curvatures of a non-degenerate curve and the focal curvatures of tangent indicatrix of the curve. Also we give the relationship between the Frenet apparatus (vectors and the curvature functions) of a curve alpha in E-n (+1) and the Frenet apparatus of tangent indicatrix alpha(T) of the curve alpha. In the main theorem of the paper, we give a characterization for a curve to be a (n-1)-spherical curve in S-n by using focal curvatures of the curve. Furtermore we give that harmonic curvature of the curve is focal curvature of the tangent indicatrix.