Progress in Invariant and Preserving Transforms for the Ratio of Collinear Points in a Desargues Affine Plane Skew Field

This paper introduces invariant transforms that preserve the ratio of either two or three points collinear in the Desargues affine plane skew field. The results given here have a clean, geometric presentation based based Desargues affine plan axiomatics and definitions with skew field properties. The main results in this paper, are (1) the ratio of two and three points is Invariant under the following transforms: Inversion, Natural Translation, Natural Dilation, Mobius Transform, in a line of Desargues affine plane. (2) parallel projection of a pair of lines in a Desargues affine plane preserves the ratio of two and three points, (3) translations in the Desargues affine plane preserve the ratio of 2 and 3 points and (4) dilation in a Desargues affine plane preserves the ratio of 2 and 3 points.