The Fundamental Theorems of Affine and Projective Geometry Revisited

TL;DR

This paper extends classical theorems in affine and projective geometry by demonstrating that under relaxed conditions, mappings preserving certain line structures are necessarily affine or projective linear, with specific minimal direction and point configurations.

Contribution

It generalizes fundamental theorems by showing that mappings with limited line-preserving properties are affine or projective linear under minimal conditions.

Findings

Injective maps with lines parallel to finitely many directions are polynomially restricted.

Five directions in 3D space suffice for affine-additivity.

n+2 fixed points in projective space imply projective-linearity.

Abstract

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.