---
tag: 323GN6K117
kind: concept
name: Angle of parallelism
msc: [51M10, 51M09]
sources:
  - url: https://en.wikipedia.org/wiki/Angle_of_parallelism
same_as:
  wikidata: Q3617405
  wikipedia: Angle of parallelism
---

## Summary

Take a line of the [[hyperbolic plane|GN4N94GQ0Z]] and a point at distance $a$ from it.
The angle of parallelism $\Pi(a)$ is the angle between the perpendicular to the line and a limiting parallel through the point.
It depends only on the distance $a$, and in curvature $-1$ it is given by $\tan(\Pi(a)/2) = e^{-a}$.
It decreases from $\pi/2$ to $0$ as $a$ grows, which measures how far hyperbolic geometry departs from Euclid's [[parallel postulate|2PC3XY0PYZ]].
The notation $\Pi(a)$ goes back to Lobachevsky.

## Explanations

### L1 human

Stand at a point $P$ at distance $a$ from a line, and drop the perpendicular to it.
Turn a ray from the perpendicular toward the line: at first the ray meets the line, and past a certain angle it never does.
That critical angle is the angle of parallelism.
In Euclidean geometry it is always a right angle, but in hyperbolic geometry it is smaller, and the farther you stand, the smaller it gets.
For example, at distance $a = \ln 3 \approx 1.1$ one gets $\tan(\Pi/2) = 1/3$, so $\Pi \approx 37^\circ$.