Progress in Invariant and Preserving Transforms for the Ratio of Co-Linear Points in the Desargues Affine Plane Skew Field

TL;DR

This paper studies invariant geometric transforms in the Desargues affine plane skew field that preserve ratios of co-linear points, providing a geometric framework based on axioms and skew field properties.

Contribution

It introduces and proves invariance of ratios under various transforms like inversion, translation, dilatation, and Möbius in the Desargues affine plane.

Findings

Ratios of two and three points are invariant under key transforms.

Parallel projection preserves point ratios in the Desargues affine plane.

Translations and dilatations preserve ratios of co-linear points.

Abstract

This paper introduces invariant transforms that preserve the ratio of either two or three co-linear points in the Desargues affine plane skew field. The results given here have a clean, geometric presentation based based Desargues affine plan axiomatic and definitions with skew field properties. The main results in this paper, are (1) ratio of two and three points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural dilatation, Mobi\"us Transform, in a line of Desargues affine plane. (2) parallel projection of a pair of lines in the Desargues affine plane preserves the ratio of two and three points, (3) translations in the Desargues affine plane preserve the ratio of 2 and 3 points and (4) dilatation in the Desargues affine plane preserve the ratio of 2 and 3 points.