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Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate. [18] He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them. [17] Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.
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]
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic.
The new axiom is Lobachevsky's parallel postulate (also known as the characteristic postulate of hyperbolic geometry): [75] Through a point not on a given line there exists (in the plane determined by this point and line) at least two lines which do not meet the given line. With this addition, the axiom system is now complete.
All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of the parallel postulate, depending on assumptions that are now recognized as essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.
Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists.
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).
However, parallel (non-crossing) pairs of lines are less restricted in hyperbolic line arrangements than in the Euclidean plane: in particular, the relation of being parallel is an equivalence relation for Euclidean lines but not for hyperbolic lines. [51] The intersection graph of the lines in a hyperbolic arrangement can be an arbitrary ...