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    Characterizations of the position vector of a surface curve in Euclidean 3-space
    (Ovidius Univ Press, 2011) Camci, Cetin; Kula, Levent; Ilarslan, Kazim
    In this paper, we give some characterizations of position vector of a unit speed curve in a regular surface M subset of E-3 which always lies in the planes spanned by {T, Z}, {T, Y} and {Y, Z}, respectively, by using (curve-surface)-frame {T,Y,Z} instead of Frenet frame {T, N, B}. We characterize such curves in terms of the geodesic curvature k(g), normal curvature k(n) and geodesic torsion t(r). Furthermore, we give some characterization for the regular surface M by using the concept of transversality of surfaces in Euclidean 3-space.
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    Characterizations of timelike slant helices in Minkowski 3-space
    (Univ Osijek, Dept Mathematics, 2014) Gok, Ismail; Nurkan, Semra Kaya; Ilarslan, Kazim; Kula, Levent; Altinok, Mesut
    In this paper, we investigate the tangent indicatrix, the principal normal indicatrix and the binormal indicatrix of a timelike curve in Minkowski 3-space E-1(3) and we construct their Frenet equations and curvature functions. Moreover, we obtain some differential equations which characterize a timelike curve to be a slant helix by using the Frenet apparatus of a spherical indicatrix of the curve. Also, related examples and their illustrations are given.
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    Harmonic curvatures and generalized helices in En
    (Pergamon-Elsevier Science Ltd, 2009) Camcı, Çetin; İlarslan, Kazım; Kula, Levent; Hacisalihoglu, H. Hilmi
    In n-dimensional Euclidean space E-n, harmonic curvatures of a non-degenerate curve defined by Ozdamar and Haci-salihoglu [Ozdamar E, Hacisalihoglu HH. A characterization of Inclined curves in Euclidean n-space. Comm Fac Sci Univ Ankara, Ser A 1 1975;24:15-23]. In this paper, we give some characterizations for a non-degenerate curve alpha to be a generalized helix by using its harmonic curvatures. Also we define the generalized Darboux vector D of a non-degenerate curve alpha in n-dimensional Euclidean space E-n and we show that the generalized Darboux vector D lies in the kernel of Frenet matrix M(s) if and only if the curve a is a generalized helix in the sense of Hayden. (C) 2007 Elsevier Ltd. All rights reserved.