non-Euclidean geometry

Hyperbolic Geometry

In hyperbolic geometry the two rays extending out in either direction from a point P and not meeting a line L are considered distinct parallels to L ; among the results of this geometry is the theorem that the sum of the angles of a triangle is less than 180°. One surprising result is that there is a finite upper limit on the area of a triangle, this maximum corresponding to a triangle all of whose sides are parallel and all of whose angles are zero. Lobachevsky's geometry is called hyperbolic because a line in the hyperbolic plane has two points at infinity, just as a hyperbola has two asymptotes. The analogy used in considering this geometry involves the lines and figures drawn on a saddleshaped surface.

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.