Hyperbolic geometry and spherical, or elliptical, geometry are two types of non-Euclidean geometry. Spherical geometry is somewhat similar to Euclidean, or plane, geometry except that it is used to determine distances and areas on the surface of a sphere instead of the flat surfaces of Euclidean geometry. Hyperbolic geometry differs from spherical geometry by its application to surfaces with a constant negative curvature, such as the curved space first introduced in Einstein's 1915 general theory of relativity.
Euclidean geometry, which dates back to 300 B.C., had a wide variety of practical applications for the Ancient Greeks who used it to survey land, design buildings and determine distances. All of these applications dealt with flat surfaces. Spherical geometry was required for navigational purposes after it was discovered that the Earth was not flat as originally thought. In the modern world, spherical geometry is relied upon by ship captains and aircraft pilots to determine the shortest distance between two points. Because spherical geometry deals with surfaces that have a constant positive curvature, the results can often be somewhat non-intuitive.
Prior to Einstein's mathematical depiction of curved space, hyperbolic geometry had no practical application because of the lack of a real-world example of a surface with a constant negative curvature. It was discovered in 1919, however, that the light coming from a distant star could be bent if it passed close enough to an object with a huge gravitational pull. Hyperbolic geometry soon began to be used to perform calculations in the new world of space, time, light and gravity described in Einstein's general theory of relativity.