Progress in Invariant and Preserving Transforms for the Ratio of Co-Linear Points in the Desargues Affine Plane Skew Field

TL;DR

The paper addresses preserving the ratio of co-linear points on lines of a Desargues affine plane endowed with a skew-field structure, seeking invariant transforms that leave these ratios unchanged. It introduces coordinate-free, geometric transforms—$j_P$, $\varphi_P$, $\delta_n$, $m$, and $\mu$—and proves their invariance properties for the 2-point and 3-point ratios defined by $r(A:B)=B^{-1}A$ and $r(A,B;C)=(B-C)^{-1}(A-C)$. It also shows that parallel projection, translations, and dilatations preserve these ratios on lines and between isomorphic lines, yielding sub-skew-fields $\mathcal{R}_2$ and $\mathcal{R}_3$ of the line skew-field. Together, these results provide a coordinate-free framework connecting affine-plane transformations with skew-field operations on lines and reinforcing invariant geometry in the Desargues setting.

Abstract

This paper introduces invariant transforms that preserve the ratio of either two or three co-linear points in the Desargues affine plane skew field. The results given here have a clean, geometric presentation based based Desargues affine plan axiomatic and definitions with skew field properties. The main results in this paper, are (1) ratio of two and three points is \emph{Invariant} under transforms: Inversion, Natural Translation, Natural dilatation, Mobiüs Transform, in a line of Desargues affine plane. (2) parallel projection of a pair of lines in the 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) dilatation in the Desargues affine plane preserve the ratio of 2 and 3 points.

Figures (12)

• Figure 1: Desargues Axioms: (a) For parallel lines $\ell^{AA'} \parallel \ell^{BB'} \parallel \ell^{CC'}$; (b) For lines which are cutting in a single point $P$, $\ell^{AA'} \cap \ell^{BB'} \cap \ell^{CC'}=P$.
• 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 $R=r(A,B;C)$.
• Figure 5: Ratio of 2-Points into parallel projection in Desargues affine plane.

Theorems & Definitions (25)

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