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

TL;DR

The paper investigates constant equiaffine curvature for curves in dual geometric settings, first establishing a robust dual-arc-length framework in D^2 and then extending to the dual Lorentz–Minkowski plane. It shows that dual-constant curvature in D^2 must be either purely real or purely dual and provides explicit real and dual component forms; in D_1^2 it offers a complete spacelike and timelike classification with concrete parametrizations. The work advances understanding of dual affine and dual Lorentzian invariants, offering complete classifications under dual symmetry groups SL(2, D) and Lorentzian isometries. These results clarify the structure of dual-space curves and supply explicit models for further geometric and applications-oriented studies.

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.

Theorems & Definitions (24)

• Definition 3.1
• Proposition 3.2
• Remark 3.3
• Example 3.4
• Definition 3.5
• Theorem 3.6
• proof
• Corollary 3.7
• Theorem 3.8
• proof