Generalization of general helices and slant helices

Abstract

In this work, we use the formal definition of $k$-slant helix cite{ali2} to obtain the intrinsic equations as well as the position vector for emph{slant-slant helices} which a generalization of emph{general helices} and emph{slant helices}. Also, we present some characterizations theorems for $k$-slant helices and derived, in general form, the intrinsic equations for such curves. Thereafter, from a Salkowski curve, anti-salkowski curve, a curve of constant precession and spherical slant helix, as examples of slant helices, we apply this method to find the parametric representation of some emph{slant-slant} helices by means of intrinsic equations. Finally, the parametric representation and the intrinsic equations of textit{Slakowski slant-slant} and textit{Anti-Slakowski slant-slant} helices have been given.

Keywords

• general helix
• slant helix
• slant-slant helix
• $k$-slant helix