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In neutral or
A ray is a limiting parallel to a ray if they are
absolute geometry
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 suf ...
, and in hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, there may be many lines parallel to a given line through a point not on line ; however, in the plane, two parallels may be closer to than all others (one in each direction of ).
Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from el, ὅριον — border).
For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal.
If, in a hyperbolic 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''.
Just as in the Euclidean case, three poi ...
, the pairs of sides are limiting parallel, then the triangle is an ideal triangle
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 sometime ...
.
Definition
A ray is a limiting parallel to a ray if they are coterminal
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
or if they lie on distinct lines not equal to the line , they do not meet, and every ray in the interior of the angle meets the ray .
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 which either is on. Then they must meet on the side of opposite to , call this point . Thus . Contradiction.See also
*horocycle
In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horospher ...
, In Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
whose normals are limiting parallels
* angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , 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 an ...
References