*(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 |