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

TL;DR

The paper addresses how the cross-ratio of four collinear points on a line of a Desargues affine plane behaves under geometric transforms. It builds an axiomatic, coordinate-free framework by leveraging the skew-field structure on each line, defines the cross-ratio $c_r$ via $c_r(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]$, and analyzes both invariant and preserving transforms. The main results establish explicit invariance of $c_r$ under invariant transforms such as inversion $j_P$, reflection $j_{-I}$, natural translation $\varphi_P$, natural dilation $\delta_n$, and Möbius transform $\mu$, as well as preservation under translations, parallel projections, and dilations, with proofs rooted in the algebraic properties of skew-fields. This work strengthens the link between axiomatic Desarguesian geometry and algebraic structures, providing algebraic tools for cross-ratio manipulation in affine/projective-geometric settings.

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, Mobiüs 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.

Figures (8)

• Figure 1: Desargues Axioms: (a) For parallel lines $\ell^{AA_1} \parallel \ell^{BB'} \parallel \ell^{CC'}$; (b) For lines which are cutting in a single point $P$, $\ell^{AA'} \cap \ell^{BB'} \cap \ell^{CC'}=P$.
• Figure 2: (a) Addition of points in a line in affine plane, (b) Multiplication of points in a line in affine plane
• Figure 3: Ilustrate the Ratio-Point, of 2-Points in a line of Desargues affine plane $R=r(A:B)=B^{-1}A$.
• Figure 4: Ratio of 3-Points in a line of Desargues affine plane $R=r(A,B;C)$.
• Figure 5: Ilustrate the Inversion of Points, in a line of Desargues affine plane $J_P(A)=P \cdot A$.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Definition 5
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• Definition 7
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• Definition 9