Constant curvature curves in dual affine and dual Lorentz-Minkowski planes

Abstract

In this paper, we first study invariants of curves parametrized by a real variable in the dual plane ${\mathbb{D}}^2$ under equiaffine transformations. We then obtain explicit equations for all curves in ${\mathbb{D}}^2$ whose equiaffine curvature is a dual constant. In particular, we prove that when the equiaffine curvature is a pure real constant, both the real and dual parts of the curve in ${\mathbb{D}}^2$ are quadratic curves. In addition, we provide a complete classification of spacelike and timelike curves parametrized by a real variable in the dual Lorentz--Minkowski plane ${\mathbb{D}}_1^2$ whose curvature is a dual constant.