On characterizations of  g n -helices with Euler angles

Abstract Curvature and torsion, in their most general forms, are defined in terms of Euler angle-based parametrization via the Cartan matrix, which leads to the Serret-Frenet equations. Euler angles are used to develop simple and general expressions for the elements of a curve with respect to the Cartan matrix. In this study, special space curves (general helix, slant helix, clad helix, etc.) are characterized with the aid of curvature and torsion equations, which are the results of the Serret-Frenet equations obtained by using the Cartan matrix created with Euler angles. Then, relations describing the transition from general helix ( g -helix) to slant helix, from slant helix ( g 2 -helix) to clad helix, from clad helix ( g 3 -helix) to g-clad helix, and similar transitions were obtained, as well as relations between the curvatures of these special curves. Finally, Illustrative examples were visualized using Wolfram Mathematica.