Generalized framed helices in Euclidean 3-space

M. Mak

In this study, we introduce framed k-slant helices from the viewpoint of singularity theory in Euclidean 3-space. Framed k-slant helices are regular or singular curves in Euclidean 3-space. Particularly, the family of framed k-slant helices encompasses a comprehensive set of curves, including framed helices, framed slant helices, and framed clad helices in the category of framed curves, as well as helices, slant helices, regular clad helices, and g-clad curves. We construct successive alternative adapted frames to define the framed k-slant helix and express its axis. We obtain a characterization of the framed (k+1)-slant helix in terms of the k-alternative adapted frame. Moreover, we discover interesting relationships between the framed k-slant helix and its generalized spherical images with respect to the adapted frame. Finally, we display appealing visualizations of the framed helix, framed slant helix, and framed clad helix as regular or smooth curves with singular points.

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 16,  supplement issue 3 (2023), pp. 59-76