A projective analogue of Napoleon's and Varignon's theorems

TL;DR

This paper introduces a projective geometry analogue of Napoleon's and Varignon's theorems, showing how certain polygon iterations lead to regularization across different geometric contexts.

Contribution

It presents a novel projective geometry theorem that generalizes classical polygon regularization results, bridging Euclidean and projective geometries.

Findings

Polygon iterations regularize shapes in projective geometry.

New theorem generalizes Napoleon's and Varignon's results.

Establishes connections between Euclidean and projective geometric regularization.

Abstract

What do Napoleon's and Varignon's theorems have in common? We claim that they both are examples of immediately regularizing natural polygon iterations in different planar geometries and present a new and analogous result in projective geometry.