Differential geometry of curves in dual space

TL;DR

This paper develops the Frenet theory for curves in dual space, defining curvature and torsion, classifying constant curvature curves, and establishing an existence theorem for curves with prescribed curvature and torsion.

Contribution

It introduces the Frenet theory in dual space, provides classification results, and proves an existence theorem for dual curves with given curvature and torsion.

Findings

Classification of dual space curves with constant curvature

Existence theorem for dual curves with prescribed curvature and torsion

Complete characterization of dual curves with constant curvature and torsion

Abstract

We introduce the Frenet theory of curves in dual space $\d^3$. After defining the curvature and the torsion of a curve, we classify all curves in dual plane with constant curvature. We also establish the fundamental theorem of existence in the theory of dual curves, proving that there is a dual curve with prescribed curvature and torsion. Finally we classify all dual curves with constant curvature and torsion.