Locating points in the pentagonal rectangular tiling of the hyperbolic plane

This paper describes an approach to locating points in the hyperbolic plane. Our technique follows the splitting of the hyperbolic plane which generates the pentagrid, i.e. the tiling of the hyperbolic plane with regular rectangular pentagons, see [2,5]. It uses also other aspects of the technique which was initiated in [2]: the spanning tree of the dual graph of the pentagrid and the coding of the situation of the pentagons by the standard Fibonacci representation of the numbers being associated of the elements of the pentagrid. We provide algorithms which, from coordinates of a point p(x(p), y(p)) in the Poincare disk allow us to obtain the number of the pentagon which contains p(x(p), y(p)).