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Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic n-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O( n ) and O(1), where O( n ) acts on the tangent space of a ...
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common perpendicular (a hyperbolic line which is perpendicular to both lines).
Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry ...
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
Angle of parallelism in hyperbolic geometry. In hyperbolic geometry, angle of parallelism () is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
The real line with the point at infinity; it is called the real projective line. In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane.