2025/03/27 by Zaka, Orgest
#30C20 #30F40 #51-XX #51A30 #51Axx #51E15 #51N25 #FOS: Mathematics #General Mathematics (math.GM)
paper · doi:10.48550/arxiv.2504.07977
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 RA4=\cr(X,B;C,D) | ∀ X ∈ ℓOI \, RB4=\cr(A,X;C,D) | ∀ X ∈ ℓOI \, RC4=\cr(A,B;X,D) | ∀ X ∈ ℓOI \ and RD4=\cr(A,B;C,X) | ∀ X ∈ ℓOI \. We disuse and examine algebraic properties for each case, related to the actions of addition and multiplication of points in ℓOI line in Desargues affine planes, which are produced by these map sets.