Below, both non-Euclidean geometries and Euclidean geometry will be compared, first as synthetic theories, then in the context of differential geometry and, lastly, in the context of group theory. The simultaneous investigation of the three geometries has made it possible to reveal the special features of each to a considerable degree and also to determine their relationships with other geometric systems. The later-discovered elliptic geometry is in some respects the opposite of hyperbolic geometry.
Hyperbolic geometry was the first geometric system distinct from Euclidean geometry, and the first more general theory (it includes Euclidean geometry as a limiting case). The major non-Euclidean geometries are hyperbolic geometry or Lobachevskii geometry and elliptic geometry or Riemann geometry - it is usually these that are meant by "non-Euclidean geometries". In the Euclidean three-dimensional space every figure can be moved in such a way that any selected point of the figure will occupy any prescribed position in addition, every figure can be rotated about any axis through any of its points. The degree of freedom of motion of figures in the Euclidean plane is characterized by the condition that every figure can be moved, without changing the distances between its points, in such a way that any selected point of the figure can be made to occupy a previously-designated position moreover, every figure can be rotated about any of its points. In the literal sense - all geometric systems distinct from Euclidean geometry usually, however, the term "non-Euclidean geometries" is reserved for geometric systems (distinct from Euclidean geometry) in which the motion of figures is defined, and this with the same degree of freedom as in Euclidean geometry.
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