Hyperbolic and elliptical geometry are the non-Euclidean alternative geometries mentioned above. The first alternative is to allow two parallels through any particular external point—or hyperbolic geometry. This studies two rays extending out in either direction from a point P, and not meeting a line L; thus, the rays are considered to be parallel to L. This also helps prove the theorem that the sum of the angles of a triangle is less than 180 degrees. It is called hyperbolic because a line in the hyperbolic plane has two points at infinity; this is similar to drawing a hyperbola that has two asymptotes.

The second alternative, called elliptical geometry, has no parallels to a given line L through an external point P. In addition, the sum of a triangle’s angles is greater than 180 degrees. Sometimes called Riemann’s geometry (who developed the idea even further; see below), it is called elliptic in general because a line in the plane of this geometry has no point at infinity (where parallels may intersect it), which is similar to an ellipse that has no asymptotes. (For more information about hyperbolas, ellipses, and asymptotes, see elsewhere in this chapter).