Comparison of Euclidean, spherical, and hyperbolic geometriesliterally any geometry that is not the same as Euclidean geometry. Although the term is frequently used to refer only to hyperbolic geometry (non-Euclidean geometry), common usage includes those few geometries (hyperbolic and spherical (non-Euclidean geometry)) that differ from but are very close to Euclidean geometry (see table (Comparison of Euclidean, spherical, and hyperbolic geometries)).

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.

It has been demonstrated by mathematics that the surface of the land and water is in its entirety a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles.

Great circles are the “straight lines” of spherical geometry. This is a consequence of the properties of a sphere, in which the shortest distances on the surface are great circle routes. Such curves are said to be “intrinsically” straight. (Note, however, that intrinsically straight and shortest are not necessarily identical, as shown in the figure-->.) Three intersecting great circle arcs form a spherical triangle (see figure-->); while a spherical triangle must be distorted to fit on another sphere with a different radius, the difference is only one of scale. In differential geometry, spherical geometry is described as the geometry of a surface with constant positive curvature.

Elliptic geometry is the term used to indicate an axiomatic formalization of spherical geometry in which each pair of antipodal points is treated as a single point. An intrinsic analytic view of spherical geometry was developed in the 19th century by the German mathematician Bernhard Riemann (Riemann, Bernhard); usually called the Riemann sphere (see figure-->), it is studied in university courses on complex analysis (analysis). Some texts call this (and therefore spherical geometry) Riemannian geometry, but this term more correctly applies to a part of differential geometry that gives a way of intrinsically describing any surface.

In 1868 the Italian mathematician Eugenio Beltrami (Beltrami, Eugenio) described a surface, called the pseudosphere, that has constant negative curvature. However, the pseudosphere is not a complete model for hyperbolic geometry, because intrinsically straight lines on the pseudosphere may intersect themselves and cannot be continued past the bounding circle (neither of which is true in hyperbolic geometry). In 1901 the German mathematician David Hilbert (Hilbert, David) proved that it is impossible to define a complete hyperbolic surface using real analytic functions (essentially, functions that can be expressed in terms of ordinary formulas). In those days, a surface always meant one defined by real analytic functions, and so the search was abandoned. However, in 1955 the Dutch mathematician Nicolaas Kuiper proved the existence of a complete hyperbolic surface, and in the 1970s the American mathematician William Thurston described the construction of a hyperbolic surface. Such a surface, as shown in the figure-->, can also be crocheted.

In the 19th century, mathematicians developed three models of hyperbolic geometry that can now be interpreted as projections (or maps) of the hyperbolic surface. Although these models all suffer from some distortion—similar to the way that flat maps distort the spherical Earth—they are useful individually and in combination as aides to understand hyperbolic geometry. In 1869–71 Beltrami and the German mathematician Felix Klein (Klein, Felix) developed the first complete model of hyperbolic geometry (and first called the geometry “hyperbolic”). In the Klein-Beltrami model (shown in the figure-->, top left), the hyperbolic surface is mapped to the interior of a circle, with geodesics in the hyperbolic surface corresponding to chords in the circle. Thus, the Klein-Beltrami model preserves “straightness” but at the cost of distorting angles. About 1880 the French mathematician Henri Poincaré (Poincaré, Henri) developed two more models. In the Poincaré disk model, the hyperbolic surface is mapped to the interior of a circular disk, with hyperbolic geodesics mapping to circular arcs (or diameters) in the disk that meet the bounding circle at right angles. In the Poincaré upper half-plane model, the hyperbolic surface is mapped onto the half-plane above the x-axis, with hyperbolic geodesics mapped to semicircles (or vertical rays) that meet the x-axis at right angles. Both Poincaré models distort distances while preserving angles as measured by tangent lines.

David W. Henderson and Daina Taimina, Experiencing Geometry: Euclidean and Non-Euclidean with History, 3rd ed. (2005), compares non-Euclidean geometries and includes directions for constructing hyperbolic surfaces. John McCleary, Geometry from a Differentiable Viewpoint (1994), emphasizes the history of the subject from Euclid's fifth (parallel) postulate and the development of the hyperbolic plane through the genesis of differential geometry. University of Minnesota, Geometry Center, Not Knot (1991), is a videotaped documentary that combines computer animations of hyperbolic spaces with a discussion that is accessible to a general audience.