CHAPTER 15

Discrete Non-Euclidean Geometry

J.J. SEIDEL

Technical University Eindhoven, Eindhoven, The Netherlands

Contents

Introduction

1. Inner product spaces

2. Spherical geometry

2.0. Summary

2.1. Spherical harmonics

2.2. Spherical codes

2.3. Spherical designs

2.4. The group case

2.5. Other types of designs

3. Elliptic geometry

3.0. Summary

3.1. The elliptic plane

3.2. Equidistant sets in elliptic -space

3.3. Regular two-graphs

3.4. Few-distance sets in

4. Hyperbolic geometry

4.0. Summary

4.1. Bolyai--Lobachevsky geometry

4.2. Harmonics in

4.3. Graphs

4.4. Integral lattices

4.5. Distance matrices

References