# General-affine invariants of plane curves and space curves

## TL;DR

This paper develops a comprehensive theory of invariants for plane and space curves under general-affine transformations, introducing new invariants, studying extremal problems, and relating to other geometric frameworks.

## Contribution

It introduces the general-affine length and curvature invariants for curves in affine geometry, and analyzes extremal problems and relations with other affine and projective theories.

## Key findings

- Defined general-affine length and curvatures for curves.
- Derived variational formulas for the length functional.
- Provided examples and connections to other geometric treatments.

## Abstract

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.