Position vectors of slant helices in Euclidean space E${}^3$

Abstract

In classical differential geometry, the problem of the determination of the position vector of an arbitrary space curve according to the intrinsic equations $\kappa =\kappa (s)$ and $\tau =\tau (s)$ (where $\kappa$ and $\tau$ are the curvature and torsion of the space curve $\psi$, respectively) is still open \cite{eisenh, lips}. However, in the case of a plane curve, helix and general helix, this problem is solved. In this paper, we solved this problem in the case of a slant helix. Also, we applied this method to find the representation of a Salkowski, anti-Salkowski curves and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.

14 pages, 3 figures

References in corpus (2)

• Curves with constant curvature ratios
• Determination of the position vectors of general helices from intrinsic equations in $\e^3$

Cited by in corpus (1)

• New special curves and their spherical indicatrices