Projective Rectangles: A New Kind of Incidence Structure

TL;DR

This paper introduces the concept of projective rectangles, a new incidence structure similar to projective planes but with differing lengths in two directions, and develops their foundational properties.

Contribution

It presents the initial theory of projective rectangles, including their incidence properties, subplanes, configurations, and connections to projective planes, offering a new geometric framework.

Findings

Defined the basic properties of projective rectangles.

Established a construction method from projective planes.

Explored partial Desargues's theorem in this context.

Abstract

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial Desargues's theorem, a construction from projective planes, and alternative formulations. In sequels we study harmonic conjugation and the graphs of lines and subplanes.