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

2022/09/06 by Orgest Zaka, Zaka, Orgest, James F. Peters +1
Engineering · Mathematics · #30C20 #30F40 #51A30 #51E15 #51N25 #Advanced Numerical Analysis Techniques #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2209.02636

openalex publication_date 2022/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Invariant under transforms: Inversion, Natural Translation, Natural dilatation, Mobiüs 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.

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