Four cross-ratio maps sets of Points and their Algebraic Structures in a line on Desargues Affine Plane

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 and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, four types of cross-ratio maps sets of points, we discussed about for each of the 4-points of cross-ratio and we will examine the algebraic properties for each case. We are constructing four cross-ratio maps sets ${\mathcal{R}}_4^A=\left\{c_r(X,B;C,D)|\forall X\in {\ell }^{OI}\right\}$, ${\mathcal{R}}_4^B=\left\{c_r(A,X;C,D)|\forall X\in {\ell }^{OI}\right\}$, ${\mathcal{R}}_4^C=\left\{c_r(A,B;X,D)|\forall X\in {\ell }^{OI}\right\}$ and ${\mathcal{R}}_4^D=\left\{c_r(A,B;C,X)|\forall X\in {\ell }^{OI}\right\}$. We disuse and examine algebraic properties for each case, related to the actions of addition and multiplication of points in ${\ell }^{OI}$ line in Desargues affine planes, which are produced by these map sets.

14 pages, 4 figures