Characterization of the Slant Helix as Successor Curve of the General Helix

TL;DR

This paper provides a comprehensive characterization of slant helices as successor curves of general helices, offering explicit solutions to natural equations and new insights into their geometric properties in three-dimensional space.

Contribution

It introduces a generic characterization of slant helices in 3D space and derives explicit arc-length parametrizations of their tangent vectors, expanding the understanding of these curves.

Findings

Explicit solutions for natural equations of an infinite series of curves.

Characterization of slant helices in terms of curvature and torsion.

Connection of slant helices to other special curves like Salkowski curves.

Abstract

In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, including notably the plane curves and general helices. This paper shows constructively how to solve the natural equations explicitly for an infinite series of curve classes. For every Frenet curve, a family of successor curves can be constructed which have the tangent of the original curve as principal normal. Helices are exactly the successor curves of plane curves and applying the successor transformation to helices leads to slant helices, a class of curves that has received considerable attention in recent years as a natural extension of the concept of general helices. The present paper gives for the first time a generic characterization of the slant helix in three-dimensional Euclidian space in terms of its curvature and torsion, and derives an explicit arc-length parametrization of its tangent vector. These results expand on and put into perspective earlier work on Salkowski curves and curves of constant precession, both of which are subclasses of the slant helix. Keywords: Curve theory; General helix; Slant helix; Curves of constant precession; Frenet equations; Frenet curves; Frenet frame; Bishop frame; Natural equations; Successor curves; Salkowski curves; Closed curves.