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Characterizations of slant helices in Euclidean 3-space. (English) Zbl 1204.53003

In Euclidean 3-space, a smooth curve γ with non zero curvature is a slant helix if the principal normal lines of γ make a constant angle with a fixed direction. Moreover, γ is a slant helix if and only if the geodesic curvature of the spherical image of the principal normal indicatrix of γ is constant [see S. Izumiya and N. Takeuchi, Turk. J. Math. 28, No. 2, 153–163 (2004; Zbl 1081.53003)].

The authors prove some sufficient and necessary conditions for a curve γ to be a slant helix by using the tangent indicatrix, the principal normal indicatrix and the binormal indicatrix of γ, as well as the vector fields of their respective Frenet frames. Then, they find the general equation of a slant helix taking into account that the tangent indicatrix of a slant helix is a spherical helix [see L. Kula and Y. Yayli, Appl. Math. Comput. 169, No. 1, 600–607 (2005; Zbl 1083.53006)] and integrating the equation of a spherical helix obtained by J. Monterde [Bol. Soc. Mat. Mex., III. Ser. 13, No. 1, 177–186 (2007; Zbl 1177.53015]. Finally, the pictures of some slant helices and their tangent and principal normal indicatrices are drawn.

MSC:
53A04Curves in Euclidean space