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

Orgest ZAKA\(^{a,∗}\), James F. Peters\(^b\)

\(^a\)Department of Mathematics-Informatics, Faculty of Economy and Agribusiness, Agricultural University of Tirana, Tirana, Albania

\(^b\)Department of Electrical & Computer Engineering, University of Manitoba, WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty of Arts and Sciences, Adiyaman University, 02040 Adiyaman, Turkey.

Correspondence should be addressed to: ozaka@ubt.edu.al , James.Peters3@umanitoba.ca

Copyright © 2024 Orgest ZAKA, James F. Peters. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 20 September 2023 Accepted: 01 October 2024; Published Online: 07 October 2024

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 axiomatics 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, Invariant and Preserving transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is 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 Preserved under transforms: parallel projection, translations and dilation’s in the Desargues affine plane.

Keywords:

Cross Ratio, Skew-Field, Desargues Affine Plane.