Absolute geometry is inconsistent with elliptic geometry : in that theory, there are no parallel lines at all, but it is a theorem of absolute geometry that parallel lines do exist.
32.
One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry ( a fundamental part of astronomy and navigation ).
33.
In elliptic geometry, an "'elliptic rectangle "'is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90?
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For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry.
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Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises : which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?
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More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point.
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It was Grossmann who emphasized the importance of a non-Euclidean geometry called Riemannian geometry ( also elliptic geometry ) to Einstein, which was a necessary step in the development of Einstein's general theory of relativity.
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Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.
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Even though elliptic geometry is not an extension of absolute geometry ( as Euclidean and hyperbolic geometry are ), there is a certain " symmetry " in the propositions of the three geometries that reflects a deeper connection which was observed by Felix Klein.
40.
Geometries that are specializations of real projective geometry, such as Euclidean geometry, elliptic geometry or conformal geometry may be complexified, thus embedding the points of the geometry in a complex projective space, but retaining the identity of the original real space as special.
