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

TL;DR

This paper investigates constant curvature curves in dual affine and dual Lorentz-Minkowski planes, deriving explicit equations and classifying spacelike and timelike curves with constant dual curvature.

Contribution

It provides explicit equations for curves with constant equiaffine curvature in dual planes and classifies dual Lorentz-Minkowski curves with constant curvature.

Findings

Quadratic form of curves with real constant curvature

Explicit equations for dual constant curvature curves

Complete classification of spacelike and timelike dual Lorentz-Minkowski curves

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 $\mathbb{D}^2_1$ whose curvature is a dual constant.