Classifying Isometries

TL;DR

This paper develops methods to classify isometries across various geometries, including Euclidean, spherical, elliptical, hyperbolic, and $ ext{L}_p$ metrics, providing a comprehensive framework for understanding geometric transformations.

Contribution

It introduces a unified approach to classify isometries in multiple geometries, extending classical Euclidean methods to more general metric spaces.

Findings

Classified all Euclidean plane isometries.

Extended classification to spherical, elliptical, and hyperbolic geometries.

Analyzed isometries in $ ext{L}_p$$ spaces for $p eq 2$.

Abstract

An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry. We then classify all isometries of the plane equipped with the $\ell_p$ metric for any $p \geq 1$ and $p = \infty$.