A Novel Solution to the Frenet-Serret Equations

TL;DR

This paper introduces a new set of equations that describe space curves using curvature and rotation angle, providing solutions that relate to the Frenet-Serret equations and addressing certain limitations in existing methods.

Contribution

It develops a novel set of equations for space curves based on curvature and rotation angle, offering an alternative to traditional Frenet-Serret equations with explicit solutions for constant angles.

Findings

Provides a solution that indirectly solves Frenet-Serret equations.

Offers explicit solutions for constant rotation angles.

Identifies limitations when the tangent aligns with coordinate axes.

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.