- Elliptic geometry
**Elliptic geometry**(sometimes known as) is aRiemannian geometry 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'sparallel 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. A simple 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 theangle s of anytriangle 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

**R**^{n+1}, that is, the points on the surface of the unit ball in n+1 dimensional space. Lines in this model aregreat circle s; 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**R**^{n+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:$d(u,\; v)\; =\; arccos\; left(frac\{u\; cdot\; v\}\; ight).$This is homogeneous 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 projective model, lines intersect exactly once. By identifying antipodal points the hyperspherical model becomes a model for the same geometry as the projective model. A notable property of the projective model is that for even dimensions, such as the plane, the geometry is nonorientable.

A model representing the same space as the hyperspherical model can be obtained by means of

stereographic projection . Let**E**^{n}represent**R**^{n}∪ {∞}, that is, n-dimensional real space extended by a single point at infinity. We may define a metric, the "chordal metric", on**E**^{n}by:$delta(u,\; v)=frac\{2\; ||u-v|\{sqrt\{(1+||u||^2)(1+||v||^2).$where u and v are any two vectors in**R**^{n}and ||*|| is the usual Euclidean norm. We also define:$delta(u,\; infty)=delta(infty,\; u)\; =\; frac\{2\}\{sqrt\{1+||u||^2.$The result is a metric space on**E**^{n}, 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:$d(u,\; v)\; =\; 2\; arcsinleft(frac\{delta(u,v)\}\{2\}\; ight).$The result is a model of elliptic geometry.The plane geometry of the hyperspherical model is

spherical geometry , where "points" are points on thesphere , and "lines" aregreat circle s 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 oflongitude that differ by 90°. These form an angle of 90° at theNorth Pole . For the third side, take theequator . 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 inEuclidean geometry .Elliptic geometry is sometimes called

Riemannian geometry , in honor ofBernhard 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, 1983

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**elliptic geometry**— noun (mathematics) a non Euclidean geometry that regards space as like a sphere and a line as like a great circle Bernhard Riemann pioneered elliptic geometry • Syn: ↑Riemannian geometry • Topics: ↑mathematics, ↑math, ↑maths … Useful english dictionary**elliptic geometry**— Non Euclidean geometry that rejects Euclid s fifth postulate (the parallel postulate) and modifies his second postulate. It is also known as Riemannian geometry, after Bernhard Riemann. It asserts that no line passing through a point not on a… … Universalium**elliptic geometry.**— See Riemannian geometry (def. 1). * * * … Universalium**elliptic geometry.**— See Riemannian geometry (def. 1) … Useful english dictionary**elliptic space**— elliptic geometry or elliptic space noun Riemannian geometry or space • • • Main Entry: ↑ellipse … Useful english dictionary**geometry**— /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… … Universalium**Elliptic curve**— In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O . An elliptic curve is in fact an abelian variety mdash; that is, it has a multiplication defined algebraically with… … Wikipedia**Elliptic complex**— In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex… … Wikipedia**Elliptic point**— In differential geometry, an elliptic point on a regular surface in R 3 is a point p at which the Gaussian curvature K ( p ) > 0 or equivalently, the principal curvatures k 1 and k 2 have the same sign … Wikipedia**non-Euclidean geometry**— geometry based upon one or more postulates that differ from those of Euclid, esp. from the postulate that only one line may be drawn through a given point parallel to a given line. [1870 75; NON + EUCLIDEAN] * * * Any theory of the nature of… … Universalium