Volumes of polytopes in spaces of constant curvature

2013/02/20 by Abrosimov, Nikolay, Mednykh, Alexander
#51M09 #52B15 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 51M20 #Secondary 51M25

paper · doi:10.48550/arxiv.1302.4919

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 H³ and S³. 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.

Related

• The volume formulas for a spherical tetrahedron
• Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths
• On volumes of hyperbolic right-angled polyhedra
• Volumes in Hyperbolic Space
• Volume and Boundary Face Area of a Regular Tetrahedron in a Constant Curvature Space