A Geometric Representation

TL;DR

This paper offers a geometric framework linking the special orthogonal group of a 3-dimensional isotropic quadratic space to projective transformations of a line, enhancing understanding of classical projective geometry concepts.

Contribution

It introduces a novel geometric representation based on inversive transformations, connecting algebraic group isomorphisms with geometric actions on the projective line.

Findings

Provides a new geometric perspective on the isomorphism between SO(3) and projective transformations.

Reinterprets classical properties like the cross ratio through this geometric lens.

Builds on previous work on inversive transformations to deepen understanding of projective line geometry.

Abstract

This article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This geometric representation depends on the theory of inversive transformations in dimension 1 as outlined in the 2021 article Projective Line Revisited by the same author. This geometric representation also provides a new perspective on some classical properties of the projective line, such as the classical cross ratio.