Volumes of non-Euclidean polyhedra

Abstract

CONTENTS §1. Natural coordinate systems 1.1. Coordinates in the plane 1.2. Cylindrical coordinates in space §2. Calculation of some auxiliary areas and volumes 2.1. Areas of sectors 2.2. Volume of a wedge §3. General properties of the volume of a tetrahedron 3.1. The volume of a tetrahedron as an analytic function of dihedral angles 3.2. Differential of volume §4. The Lobachevskii formula 4.1. The Lobachevskii function 4.2. Birectangular tetrahedra 4.3. Volume of a birectangular tetrahedron in Lobachevskii space 4.4. Volume of a spherical birectangular tetrahedron 4.5. Volumes of truncated birectangular tetrahedra and the Lambert cube §5. Volumes of polyhedra with vertices at infinity 5.1. Volume of an ideal tetrahedron 5.2. Volume of a tetrahedron with three vertices at infinity 5.3. Volume of a pyramid with vertex at infinity References