A Third-Order Tangent Characterization of Helices under Curvature-Based Reparametrization

Abstract

In this study, space curves are investigated under a curvature-based reparametrization. Within this framework, it is shown that the Frenet–Serret system for generalized helices can be reduced to a simpler form, and that the tangent vector can be expressed through a third-order linear differential equation. Furthermore, this approach is shown to provide a characterization of helices. An associated energy conservation is established, and the results are interpreted via the spherical image of the tangent indicatrix. In addition, a functional measuring the deviation from the helical structure is introduced, and its behavior is analyzed numerically on selected example curves and visualized through the corresponding graphs.