Defining relations for reflections. I

TL;DR

This paper introduces simple generators and relations for classical isometry groups, enabling efficient geometric calculations and providing new presentations for groups like Euclidean plane, 2-sphere, and SO(3).

Contribution

It presents novel, elegant presentations of classical isometry groups using generators and relations closely tied to geometry.

Findings

Efficient presentations for Euclidean plane and 2-sphere isometry groups.

New relations for groups SO(3) and O(n).

Facilitates fast geometric computations.

Abstract

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and relations closely related to geometry. They allow to make fast and efficient geometric calculations. In this paper simple presentations of the isometry groups of Euclidean plane, 2-sphere, the real projective plane and groups SO(3), O(n) are introduced.