Partitions of a hyperbolic plane of positive curvature by correct horocyclic $n$-trapezes

L. N. Romakina

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 376-416

Résumé

A hyperbolic plane $\widehat{H}$ of positive curvature is realized on the external domain whith respect to the oval curve of the projective plane $P_2$, i.e. on the ideal domain of the Lobachevskii plane. In works of the author the first partitions of the plane $\widehat{H}$ are constructed. Among them there are series of the normal, but not monohedral partitions and the series of the monohedral partitions which are not the normal. In this work the series of the first normal monohedral partitions of the plane $\widehat{H}$ are constructed. One of topological differences of the plane $\widehat{H}$ from the Lobachevskii plane is in the following fact. No line of the plane $\widehat{H}$ partitions the plane (the set of Betti numbers for the plane $\widehat{H}$: $\beta_0 = 1$, $\beta_1 = 1$, for the plane $\Lambda^2$: $\beta_0 = 1$, $\beta_1 = 0$). Therefore the main known methods of a construction of partitions of the Lobachevskii plane can not be applied in partitions of the plane $\widehat{H}$. As an exception it is possible to consider the tiling scheme of the plane $\Lambda^2$ offered by the Hungarian mathematician K. Beretsky. In present work Beretsky's scheme is adapted for the plane $\widehat{H}$. On the basis of this scheme the normal monohedral partitions the plane $\widehat{H}$ with one remote parabolic line are constructed. The cells of the constructed partitions are the correct horocyclic $n$-trapezes. They are in detail investigated in this work. The correct horocyclic $n$-trapeze called the $(n+3)$-hedral which contain two congruent edges on the parallel hyperbolic lines. The other edges of $(n+3)$-hedral are the congruent elliptic segments. One of them serves as an internal chord of some horocycle $\omega$, and other $n$ segments are the internal chords of the concentric with $\omega$ horocycle. For research of the cells of partitions in present work the orthogonal horocyclic coordinate system is entered. Auxiliary formulas of the areas of some figures of the plane $\widehat{H}$ are received. It is proved that the area of the correct horocyclic $n$-trapeze can be expressed by means of the function $\widetilde{\alpha}$ of a quasiparallelism angle entered by the author on the plane $\widehat{H}$. The length of the side edge no depend from the length of elliptic edges and is equal to $\rho \ln n$, where $\rho$ is the radius of curvature of the plane $\widehat{H}$. Bibliography: 19 titles.

Détail

Keywords: a hyperbolic plane $\widehat{H}$ of positive curvature, a correct horocyclic $n$-trapeze, normal monohedral partitions of the plane $\widehat{H}$, Beretsky's scheme.