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°.
Two dimensions[edit]
Elliptic plane[edit]
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.[3]
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".[4]: 82 This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.
Higher-dimensional spaces[edit]
Hyperspherical model[edit]
The hyperspherical model is the generalization of the spherical model to higher dimensions. The points of n-dimensional elliptic space are the pairs of unit vectors (x, −x) in Rn+1, that is, pairs of antipodal points on the surface of the unit ball in (n + 1)-dimensional space (the n-dimensional hypersphere). Lines in this model are great circles, i.e., intersections of the hypersphere with flat hypersurfaces of dimension n passing through the origin.
Projective elliptic geometry[edit]
In the projective model of elliptic geometry, the points of n-dimensional real projective space are used as points of the model. This models an abstract elliptic geometry that is also known as projective geometry.
The points of n-dimensional projective space can be identified with lines through the origin in (n + 1)-dimensional space, and can be represented non-uniquely by nonzero vectors in Rn+1, with the understanding that u and λu, for any non-zero scalar λ, represent the same point. Distance is defined using the metric
Self-consistency[edit]
Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry.
Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.[8] (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[9]) It therefore follows that elementary elliptic geometry is also self-consistent and complete.