Abstract. We consider a curve α = α(s) in Minkowski 3-space E 3 1 and denote by {T, N, B} the Frenet frame of α. We say that α is a slant helix if there exists a fixed direction U of E 3 1 such that the function ⟨N(s), U ⟩ is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of α. Finally, we discuss the tangent and binormal indicatrices of slant curves, proving that they are helices in E 3 1 .

In this paper, position vector of a slant helix with respect to standard frame in Euclidean space E 3 is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine a position vector of an arbitrary slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of a Salkowski curve, anti-Salkowski curve and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.

In this paper, position vector of a time-like slant helix with respect to standard frame of Minkowski space E 3 1 is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary timelike slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the representation of a time-like Salkowski and time-like anti-Salkowski curves as examples of a slant helices, by means of intrinsic equations. Moreover, we present some new characterizations of slant helices and illustrate some examples of our main results.

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