A Novel Solution to the Frenet-Serret Equations

Abstract

A set of equations is developed to describe a curve in space given the curvature $\kappa$ and the angle of rotation $\theta$ of the osculating plane. The set of equations has a solution (in terms of $\kappa$ and $\theta$) that indirectly solves the Frenet-Serret equations, with a unique value of $\theta$ for each specified value of $\tau$. Explicit solutions can be generated for constant $\theta$. The equations break down when the tangent vector aligns to one of the unit coordinate vectors, requiring a reorientation of the local coordinate system.