Volumes of Polytopes in Spaces of Constant Curvature

Abstract

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.

To a question on the existence of regular mosaics on a hyperbolic plane of positive curvature

“…The second case does not lead to contradictions. Let us continue our reasonings, accepting the condition (2).…”

“…Later this theory was developed in works of many authors (see, for instance, [5,7,9,19,21,42,43]). Profound survey on this topic is presented in papers [2,21,44].…”

“…where S 2 .F / or S O H .F / is the area of the figure F calculated in geometry of the plane 2 or O H respectively (see [33,36]). In this paper due to the theorem on the area of a generalized polygon without parabolic edges (see [38,…”

Delaunay Triangulations of Degenerate Point Sets

“…By the normal perturbation definition, all the x i , y i are close enough to 0, so the sign of ∆(j) is determined only by the sum of its first-order terms, as long as they are not too close to zero, as it is in our case (with probability almost 1). 1 After the evaluation of the determinants and the exclusion of terms of second or higher order, we get the following system:…”

Hyperbolic Tetrahedron: Volume Calculation with Application to the Proof of the Schläfli Formula

“…Известно много работ, посвященных вычислению объема тетраэдра в пространстве Лобачевского, см. литературу, например, в работах [1] и [2]. В наших докладах на конференциях Дни геометрии в Новосибирске (27.08 31.08.2013) (см.…”