Hyperbolic spaces are a type of non-Euclidean geometry characterized by a constant negative curvature. Unlike flat or spherical geometries, hyperbolic spaces allow for unique properties, such as the fact that the angles of a triangle sum to less than 180 degrees. This makes them useful in various fields, including mathematics, physics, and computer science. One common model of hyperbolic space is the Poincaré disk model, where the entire hyperbolic plane is represented within a circular disk. In this model, lines appear as arcs that intersect the boundary of the disk at right angles. Hyperbolic spaces have applications in topology, group theory, and even in understanding the structure of the universe.