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Limiting parallel

Geometrical term

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated May 14, 2026
Image credit is listed at the end of this article.

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

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 Greek: 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, the pairs of sides are limiting parallel, then the triangle is an ideal triangle.

01Definition

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.

The ray Aa is a limiting parallel to Bb, written:
The ray Aa is a limiting parallel to Bb, written:

02Properties

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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Sources and credits

This article is adapted from the Wikipedia article “Limiting parallel”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Horocycle

In hyperbolic geometry, a horocycle, sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics (normals) through a point on a horocycle are limiting parallel, and all converge asymptotically to a single ideal point called the centre of the horocycle. In some models of hyperbolic geometry, it looks like the two "ends" of a horocycle get closer and closer to each other and closer to its centre, but this is not true; the two "ends" of a horocycle get further and further away from each other and stay at an infinite distance off its centre.

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.

Curve

In mathematics, a curve 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.