Invariant and Preserving Transforms for Cross Ratio of 4-Points in a line on Desargues Affine Plane

TL;DR

This paper explores invariant and preserving transforms of the cross ratio of four points on a line within Desargues affine planes, establishing their properties and relations to affine plane axioms and skew field structures.

Contribution

It introduces a detailed study of invariant and preserving transforms of cross ratios in Desargues affine planes, with formal axiomatic foundations and new results.

Findings

Cross ratio is invariant under inversion, natural translation, dilation, and Möbius transforms.

Cross ratio is preserved under parallel projection, translations, and dilations.

The paper formalizes these transforms within the axiomatic framework of Desargues affine planes.

Abstract

This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic's and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, \emph{Invariant} and \emph{Preserving} transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural Dilation, Mobi\"us Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is \emph{Preserved} under transforms: parallel projection, translations and dilation's in the Desargues affine plane.