![]() any two points of the plane not on a given straight line can be joined within the plane by a continuous curve that does not cut the straight line (the topological model of the elliptic plane is the projective plane). In addition, in Euclidean and hyperbolic geometries every straight line in a given plane divides the plane into two parts in elliptic geometry this is not true, i.e. similar to the order in the set of real numbers in elliptic geometry, the points of a straight line are cyclically ordered, like the points on a circle. Namely, in Euclidean and hyperbolic geometries, the order of the points on a straight line is linear, i.e. The relevant axioms of these geometries in which the differences occur are those that determine the order relations among the geometric elements. Thus, the axiom system underlying elliptic geometry must be different from that of Euclidean geometry not only in the replacement of one axiom - the parallelism axiom - but also in part of the other axioms. This axiom contradicts the system of axioms of Euclidean geometry with the parallelism axiom excluded. The axiom adopted in elliptic geometry is: Every straight line coplanar with another straight line intersects the latter. Which does not intersect it in hyperbolic geometry it is assumed that there are several such lines (and it can then be proved that there are infinitely many). There passes exactly one straight line coplanar with $ a $ Recall that according to the latter, through any point not lying on a given straight line $ a $ Hyperbolic geometry is based on the same axioms as Euclidean geometry, with the exception of the parallelism axiom. Non-Euclidean geometries as synthetic theories. 3 Non-Euclidean geometries in a group-theoretic context. ![]() 2 Non-Euclidean geometries in a differential-geometric context.1 Non-Euclidean geometries as synthetic theories.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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