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

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.