Generalized Frenet frames and frame sequences of singular space curves

Abstract

The classical Frenet frame is defined by a concrete construction from the tangent, principal normal, and binormal vectors of a regular space curve. However, this construction breaks down at singular points and at points where the curvature vanishes. Motivated by this observation, we reconsider the Frenet frame from an axiomatic viewpoint and identify the fundamental properties that characterize it independently of its classical construction. Based on the theory of frontals on the unit sphere and Legendre duality, we introduce a generalized Frenet frame for singular space curves. Furthermore, we introduce the notion of a frame sequence, which gives rise to a family of $k$th-order Frenet frames and their corresponding curvatures and torsions, indexed by $k \in \mathbb{Z}$. This viewpoint provides a unified framework encompassing the Frenet and Bishop frames of space curves as well as the evolute-involute correspondence for spherical frontals. Moreover, explicit recursive formulas are derived, revealing that the curvatures and torsions at each level encode, respectively, the magnitude and rotational behavior of the invariants at the preceding level.