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

Orgest Zaka; James F. Peters

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arXiv · 2211.16055

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

Orgest Zaka·James F. Peters

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

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BibTeX RIS

Orgest Zaka, James F. Peters. 2022-11-29. Invariant and Preserving Transforms for Cross Ratio of 4-Points in a line on Desargues Affine Plane. https://arxiv.org/abs/2211.16055

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