In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line

## Contents

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.

Limiting parallels may form two, or three sides of a limit triangle.

## Definition

A ray

## Properties

Distinct lines carrying limiting parallel rays do not meet.

## Proof

Suppose that the lines carrying distinct parallel rays met. By definition the cannot meet on the side of