Differential geometry of general affine plane curves

TL;DR

This paper explores the affine geometry of plane curves, establishing invariants like curvature and signature, and classifies constant curvature curves in affine space.

Contribution

It introduces minimal order and Frenet moving frames for affine plane curves and proves curvature and signature are complete invariants.

Findings

Curvature and signature fully classify regular affine plane curves.

Complete classification of constant curvature curves in affine space.

Development of minimal order and Frenet moving frames for affine curves.

Abstract

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature and signature are the complete invariants of regular curves. As application we give a complete classification of constant curvature curves in $A^2$.