Invariant and Preserving Transforms for Cross Ratio of 4-Points 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'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, Invariant and Preserving transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is 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 Preserved under transforms: parallel projection, translations and dilation's in the Desargues affine plane.