Some characterizations of dual curves in dual 3-space $ \mathbb{D}^{3} $

Abstract

In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is a non-constant linear function of its dual arc length parameter. Thereafter, a dual differential equation of third order is constructed for every dual curve. Then, several well-known characterizations of dual spherical, normal and rectifying curves are consequences of this differential equation. Finally, we prove a simple new characterization of dual spherical curves in terms of the Darboux vector.