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

Orgest Zaka; James Peters

doi:10.5269/bspm.68800

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

Advances in Desargues Affine Plane Transforms

Orgest Zaka Department of Mathematics - Informatics Faculty of Economy and Agribusiness Agricultural University of Tirana, Tirana, Albania. Kamza Hill, SH1, Tirana 1000, Building A, Office 232. https://orcid.org/0000-0001-8431-8347
James Peters Department of Electrical \& Computer Engineering, Computational Intelligence Laboratory, University of Manitoba, Winnipeg, Manitoba, R3T 5V6, Canada https://orcid.org/0000-0002-1026-4638

DOI: https://doi.org/10.5269/bspm.68800

Resumen

This paper introduces invariant transforms that preserve the ratio of either two or three points collinear in the Desargues affine plane skew field.
The results given here have a clean, geometric presentation based based Desargues affine plan axiomatics 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 Dilation,
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) dilation in the Desargues affine plane preserve the ratio of 2 and 3 points.