Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. As in Euclidean geometry, where ancient Greek mathematicians used a compass and idealized ruler for constructions of lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry. Models of hyperbolic geometry There are a couple of models for hyperbolic geometry, which can make performing and visualizing constructions easier. Portions of the hyperbolic plane can be placed onto a pseudosphere and maintain angles and hyperbolic distances, as well as be bent around the pseudosphere and still keep its properties. However, not the entire hyperbolic plane can be placed onto the pseudosphere as a model, only a portion of the hyperbolic plane. The entire hyperbolic plane can also be placed on a Poincaré disk and maintain its angles. However, the lines will turn into circular arcs, which warps them. Tools In hyperbolic geometry, one can use the standard ruler and compass that is often used in Euclidean plane geometry. However, there are a variety of compasses and rulers developed for hyperbolic constructions. A hypercompass can be used to construct a hypercycle given the central line and radius. A horocompass can be used to construct a horocycle through a specific point if the diameter and direction are also provided. Both of these also require a straight edge, like the standard ruler. When doing constructions in hyperbolic geometry, as long as you are using the proper ruler for the construction, the three compasses (meaning the horocompass, hypercompass, and the standard compass) can all perform the same constructions. A parallel ruler can be used to draw a line through a given point A and parallel to a given ray a. For any two lines, a hyperbolic ruler can be used to construct a line that is parallel to the first line and perpendicular to the second. A few notes on the uses of rulers are: A parallel ruler can be used to construct anything that a standard ruler and the three rulers can also construct A parallel ruler can act as a ruler in Euclidean geometry A hyperbolic ruler cannot perform Euclidean