The equidistant involution of the hyperbolic plane and two models of the Euclidean plane geometry (Q375377)

In this article, the author presents two models of Euclidean geometry in the hyperbolic plane. Of particular note is a bounded disk model of the Euclidean plane in the hyperbolic plane. The motivation for this work is the classical Poincaré disk model, which is the reverse: a bounded disk model of the hyperbolic plane in the Euclidean plane. Moreover, the hyperbolic isometries of the Poincaré disk are described in simple Euclidean terms, as inversions in a circle. To play a role analogous to Euclidean inversion, the author defines a hyperbolic involution corresponding to an equidistant curve and a point of its base line that keeps a certain subset of the equidistant curves invariant. The article concludes by demonstrating that the constructed model and the Euclidean plane are connected in a manner similar to that between the Poincaré and Beltrami-Cayley-Klein models of hyperbolic geometry.