Inversion with respect to a hypercycle of a hyperbolic plane of positive curvature

Abstract

Inversion with respect to a hypercycle of a hyperbolic plane $${\widehat{H}}$$ of positive curvature is investigated. The plane $${\widehat{H}}$$ is the projective Cayley–Klein model of two-dimensional de Sitter’s space. One of four analogs of a Euclidean circle on the plane $${\widehat{H}}$$ is a hypercycle. The formulae of inversion with respect to the hypercycle in a canonical frame of the first type are derived. The main properties of this inversion are proved.