Elliptic geometry

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Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p.

Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p. In elliptic geometry, there are no parallel lines at all. The best way to picture this is to look at a globe. The lines of longitude are exactly next to each other, yet they eventually intersect. Elliptic geometry has other unusual properties. For example, the sum of the angles of any triangle is always greater than 180°.

Models of elliptic geometry

Models of elliptic geometry include the hyperspherical model, the projective model, and the stereographic model.

In the hyperspherical model, the points of n-dimensional elliptic space are the unit vectors in Rⁿ⁺¹, that is, the points on the surface of the unit ball in n+1 dimensional space. Lines in this model are great circles; intersections of the ball with hypersurface subspaces, meaning subspaces of dimension n.

In the projective model, the points of n-dimensional real projective space are used as points of the model. 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 can be defined using the metric

<math>d(u, v) = \arccos (\frac{u \cdot v}{||u||\ ||v||}).</math>

This is homogenous in each variable, with d(λu, μv) = d(u, v) if λ and μ are non-zero scalars, and so it defines a distance on the points of projective space.

The two models represent different geometries; in the hyperspherical model, two distinct lines intersect exactly twice, at antipodal points, and in the elliptic model, lines intersect exactly once. By identifying antipodal points the hyperspherical model becomes a model for the same geometry as the elliptic model.

A model representing the same space as the hyperspherical model can be obtained by means of stereographic projection. Let Eⁿ represent Rⁿ ∪ {∞}, that is, n-dimensional real space extended by a single point at infinity. We may define a metric, the chordal metric, on Eⁿ by

<math>\delta(u, v) = \frac{2 ||u-v||}{\sqrt{(1+||u||^2)(1+||v||^2)}}.</math>

where u and v are any two vectors in Rⁿ and ||*|| is the usual Euclidean norm. We also define

<math>\delta(u, \infty)=\delta(\infty, u) = \frac{2}{\sqrt{1+||u||^2}}.</math>

The result is a metric space on Eⁿ, which represents the distance along a chord of the corresponding points on the hyperspherical model, which it maps bijectively to by stereographic projection. To obtain a model of elliptic geometry, we define another metric

<math>d(u, v) = 2 \arcsin(\frac{\delta(u,v)}{2}).</math>

The result is a model of elliptic geometry.

The plane geometry of the hyperspherical model is spherical geometry, where points are points on the sphere, and lines are great circles through those points. On the sphere, such as the surface of the Earth, it is easy to give an example of a triangle that requires more than 180°: For two of the sides, take lines of longitude that differ by 90°. These form an angle of 90° at the North Pole. For the third side, take the equator. The angle of any longitude line makes with the equator is again 90°. This gives us a triangle with an angle sum of 270°, which would be impossible in Euclidean geometry.

Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, 1983de:Elliptische Geometrie it:Geometria ellittica ja:楕円幾何学