---
tag: 1E8WCHSAE2
kind: concept
name: Horocycle
msc: [51M10, 53A35]
sources:
  - url: https://en.wikipedia.org/wiki/Horocycle
same_as:
  wikidata: Q2246795
  wikipedia: Horocycle
---

## Summary

A horocycle is a curve of the [[hyperbolic plane|GN4N94GQ0Z]] whose normal geodesics all run to the same point at infinity.
It is the limit of circles through a fixed point with a fixed tangent there, as the radius tends to infinity.
In the [[Poincaré disk model|85MM2EA0TK]] it is a Euclidean circle inside the disc tangent to the boundary circle, without the point of tangency.
Horocycles are not geodesics: they have constant geodesic curvature $1$ in curvature $-1$.

## Explanations

### L1 human

In Euclidean geometry, a circle of huge radius looks almost like a straight line.
In hyperbolic geometry this fails: as the radius grows, circles tend to a new kind of curve, the horocycle, which still bends.
All the radii of a horocycle, the geodesics that cross it at right angles, head toward one common point at infinity, its centre.
In the upper half-plane picture, the horizontal line $y = 1$ is a horocycle: the vertical half-lines cross it at right angles and all go up to the same point at infinity.