Limiting parallel

Last updated August 11, 2019

In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line $l$ through a point $P$ not on line $R$ ; however, in the plane, two parallels may be closer to $l$ than all others (one in each direction of $R$ ).

Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not sufficient as a basis of Euclidean geometry, other systems, such as Hilbert's axioms without the parallel axiom, are used. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.

Hyperbolic geometry Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Greek is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal.

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects $a$ , $b$ , and $c$ :

If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle.

In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.

Definition

A ray $Aa$ is a limiting parallel to a ray $Bb$ if they are coterminal or if they lie on distinct lines not equal to the line $AB$ , they do not meet, and every ray in the interior of the angle $BAa$ meets the ray $Bb$ .^[1]

Properties

Distinct lines carrying limiting parallel rays do not meet.

Proof

Suppose that the lines carrying distinct parallel rays met. By definition they cannot meet on the side of $AB$ which either $a$ is on. Then they must meet on the side of $AB$ opposite to $a$ , call this point $C$ . Thus $\angle CAB+\angle CBA<2{\text{ right angles}}\Rightarrow \angle aAB+\angle bBA>2{\text{ right angles}}$ . Contradiction.

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References

↑ Hartshorne, Robin (2000). Geometry: Euclid and beyond (Corr. 2nd print. ed.). New York, NY [u.a.]: Springer. ISBN 978-0-387-98650-0.

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[1] Hartshorne, Robin (2000). Geometry: Euclid and beyond (Corr. 2nd print. ed.). New York, NY [u.a.]: Springer. ISBN 978-0-387-98650-0.