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

In Euclidean 3-space, a smooth curve $\gamma$ with non zero curvature is a slant helix if the principal normal lines of $\gamma$ make a constant angle with a fixed direction. Moreover, $\gamma$ is a slant helix if and only if the geodesic curvature of the spherical image of the principal normal indicatrix of $\gamma$ 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 $\gamma$ to be a slant helix by using the tangent indicatrix, the principal normal indicatrix and the binormal indicatrix of $\gamma$, 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:
 53A04 Curves in Euclidean space