Volumes of polytopes in spaces of constant curvature

TL;DR

This paper reviews volume calculations for polyhedra in Euclidean, spherical, and hyperbolic spaces, proves formulas for tetrahedron volumes, and explores hyperbolic quadrilaterals, providing new formulas and solutions to classical problems.

Contribution

It introduces new volume formulas for hyperbolic tetrahedra and quadrilaterals, and addresses the Seidel problem in non-Euclidean geometry.

Findings

Proved Sforza formula for hyperbolic and spherical tetrahedra

Derived hyperbolic Brahmagupta-type formulas for quadrilaterals

Provided solutions to Seidel problem on non-Euclidean tetrahedron volume

Abstract

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a solution for Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find a few versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.