Elliptic geometry

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry. Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, the perpendiculars on one side all intersect at a single point called the absolute pole of that line. The perpendiculars on the other side also intersect at a point. However, unlike in spherical geometry, the poles on either side are the same. This is because there are no antipodal points in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the 'points' in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points. Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.:89 The distance between a pair of points is proportional to the angle between their absolute polars.:101 As explained by H. S. M. Coxeter The elliptic plane is the real projective plane provided with a metric: Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it. With O the center of the hemisphere, a point P in σ determines a line OP intersecting the hemisphere, and any line L ⊂ σ determines a plane OL which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of σ corresponds to this plane; instead a line at infinity is appended to σ. As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed. Given P and Q in σ, the elliptic distance between them is the measure of the angle POQ, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote 'On the definition of distance'.:82 This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.