<P> Along with the Klein model and the Poincaré half - space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry . It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami . </P> <P> The Poincaré ball model is the similar model for 3 or n - dimensional hyperbolic geometry in which the points of the geometry are in the n - dimensional unit ball . </P> <P> Hyperbolic Straight lines consist of all arcs of Euclidean circles contained within the disk that are orthogonal to the boundary of the disk, plus all diameters of the disk . </P> <P> The unique hyperbolic line through two points P and Q not on a diameter of the boundary circle can be constructed by: </P>

A line in hyperbolic geometry is an arc that intersects the boundary