Polarit\'es d\'efinies par un triangle

TL;DR

This paper explores various types of polarities in projective geometry, focusing on a new polarity defined by a triangle, and demonstrates their equivalence, linking concepts across projective, linear, and convex geometry.

Contribution

It introduces a new triangle-based polarity and proves its equivalence with classical polarities, unifying different geometric concepts.

Findings

All four polarities coincide for a triangle.

The triangle polarity relates to duality in projective frames.

Connections between projective, linear, and convex geometry are highlighted.

Abstract

A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an opportunity to review nice concepts of projective geometry, linear algebra and convex geometry.