Cross Ratio Geometry Advances for Four Co-Linear Points in the Desargues Affine Plane-Skew Field

TL;DR

The paper addresses the problem of defining and analyzing the cross-ratio of four collinear points within a Desargues affine plane over a skew field, without relying on coordinates. It develops a coordinate-free framework by treating line points as elements of a skew field and formulating cross-ratio through skew-field operations, notably $c_r(A,B;C,D)=\big[(A-D)^{-1}(B-D)\big]\big[(B-C)^{-1}(A-C)\big]$. Key contributions include establishing the skew-field structure on lines via Desarguesian axioms, detailing ratio maps $r(A:B)$ and $r(A,B;C)$ and their sub-skew-field structures, and deriving numerous identities and special cases for the cross-ratio (including behavior under inverses, infinity, and center/centralizer conditions). The results provide a robust, coordinate-free algebraic treatment of cross-ratio in Desargues affine planes, illuminating the interplay between affine axioms and skew-field geometry with potential implications for noncommutative projective-type constructions.

Abstract

This paper introduces advances in the geometry of the cross ratio of four co-linear points in in the Desargues affine plane. The cross-ratio of co-linear points of a skew field in the Desargues affine plane. The results given here have a clean rendition, based on Desargues affine plane axiomatics, skew field properties and the addition and multiplication of planar co-linear points.

Figures (4)

• 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 $P$, $\ell^{AA_1} \cap \ell^{BB'} \cap \ell^{CC'}=P$.
• 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 $R=r(A,B;C)$.

Theorems & Definitions (20)

• Definition 1
• Example 1
• Proposition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Theorem 1
• Theorem 2