On the area of a trihedral on a hyperbolic plane of positive curvature

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

“…In Theorem 2 we prove that there is no regular mosaic on the plane O H . The simple proof of this fundamental fact is based on the theorem about the polygon area in the plane O H , see Theorem 3.1 in [38]. Since various stages of the proof of Theorem 3.1 have been published in three papers, we provide its short sketch in adjuvant Section 2.…”

“…The foundation of the areas theory of the plane O H is set in works [33,36,38,39]. The notion of the area is introduced such that the finite figures of the plane O H possess the real positive areas.…”

“…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,…”

“…We prove Theorem 1 in three stages. At first, in paper [33], we choose an orthogonal hypercyclic coordinate system on the plane O H and prove the following formula for the calculation of the area of a rectangular trihedral with an elliptic (hyperbolic) cathetus of length a (b)…”

Объем конечного ортогонального h-конуса в гиперболическом пространстве положительной кривизны

“…Например, согласно Теореме 3.1 из [5] и Теореме 3 из [20] справедлива следующая теорема. При вычислении в пространстве H 3 объемов пирамид с гиперболи-ческим основанием могут быть полезны формулы площадей n-ребер-ников различных видов в гиперболической плоскости положительной кривизны из работ [20], [19], [21], [12].…”