In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180°. Riemann's geometry is called elliptic because a line in the plane described by this geometry has no point at infinity, where parallels may intersect it, just as an ellipse has no asymptotes. An idea of the geometry on such a plane is obtained by considering the geometry on the surface of a sphere, which is a special case of an ellipsoid. The shortest distance between two points on a sphere is not a straight line but an arc of a great circle (a circle dividing the sphere exactly in half). Since any two great circles always meet (in not one but two points, on opposite sides of the sphere), no parallel lines are possible. The angles of a triangle formed by arcs of three great circles always add up to more than 180°, as can be seen by considering such a triangle on the earth's surface bounded by a portion of the equator and two meridians of longitude connecting its end points to one of the poles (the two angles at the equator are each 90°, so the amount by which the sum of the angles exceeds 180° is determined by the angle at which the meridians meet at the pole).
Sections in this article:
The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.