Horocycle

In

.

In

${\displaystyle -1,}$ the curves of constant curvature come in four types: geodesics with curvature ${\displaystyle \kappa =0,}$ hypercycles with curvature ${\displaystyle 0<|\kappa |<1,}$ horocycles with curvature ${\displaystyle |\kappa |=1,}$ and circles with curvature ${\displaystyle |\kappa |>1.}$

Any two horocycles are congruent, and can be superimposed by an isometry (translation and rotation) of the hyperbolic plane.

A horocycle can also be described as the limit of the circles that share a tangent at a given point, as their radii tend to infinity, or as the limit of hypercycles tangent at the point as the distances from their axes tends to infinity.

Two horocycles with the same centre are called

. As for conceptric circles, any geodesic perpendicular to a horocycle is also perpendicular to every concentric horocycle.

Properties

• Through every pair of points there are 2 horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the segment between them.
• No three points of a horocycle are on a line, circle or hypercycle.
• All horocycles are congruent. (Even concentric horocycles are congruent to each other)
• A straight line, circle, hypercycle, or other horocycle cuts a horocycle in at most two points.
• The
  perpendicular bisector of a chord of a horocycle is a normal of that horocycle and the bisector bisects the arc subtended by the chord and is an axis of symmetry
  of that horocycle.
• The length of an arc of a horocycle between two points is:

longer than the length of the line segment between those two points,

longer than the length of the arc of a hypercycle between those two points and

shorter than the length of any circle arc between those two points.

• The distance from a horocycle to its centre is infinite, and while in some models of hyperbolic geometry it looks like the two "ends" of a horocycle get closer and closer together and closer to its centre, this is not true; the two "ends" of a horocycle get further and further away from each other.
• A regular apeirogon is circumscribed by either a horocycle or a hypercycle.
• If C is the centre of a horocycle and A and B are points on the horocycle then the angles CAB and CBA are equal.^[1]
• The area of a sector of a horocycle (the area between two radii and the horocycle) is finite.^[2]

Standardized Gaussian curvature

When the hyperbolic plane has the standardized Gaussian curvature K of −1:

• The length s of an arc of a horocycle between two points is:
  $${\displaystyle s=2\sinh \left({\frac {1}{2}}d\right)={\sqrt {2(\cosh d-1)}}}$$

  
  where d is the distance between the two points, and sinh and cosh are hyperbolic functions.^[3]
• The length of an arc of a horocycle such that the tangent at one extremity is limiting parallel to the radius through the other extremity is 1.^[4] the area enclosed between this horocycle and the radii is 1.^[5]
• The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e : 1.^[6]

Representations in models of hyperbolic geometry

Poincaré disk model

In the Poincaré disk model of the hyperbolic plane, horocycles are represented by circles tangent to the boundary circle; the centre of the horocycle is the ideal point where the horocycle touches the boundary circle.

The

where both points are inside the circle.

Poincaré half-plane model

In the Poincaré half-plane model, horocycles are represented by circles tangent to the boundary line, in which case their centre is the ideal point where the circle touches the boundary line.

When the centre of the horocycle is the ideal point at ${\displaystyle y=\infty }$ then the horocycle is a line parallel to the boundary line.

The

.

Hyperboloid model

In the hyperboloid model they are represented by intersections of the hyperboloid with planes whose normal lies on the asymptotic cone (i.e., is a null vector in three-dimensional Minkowski space.)

Metric

If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of geodesic curvature 1 at every point.

Horocycle flow

Every horocycle is the orbit of a

of the hyperbolic plane. This flow is called the horocycle flow of the hyperbolic plane.

Identifying the unit tangent bundle with the group

PSL(2,R)

, the horocycle flow is given by the right-action of the unipotent subgroup ${\displaystyle U=\{u_{t},\,t\in \mathbb {R} \}}$, where:

$${\displaystyle u_{t}=\pm \left({\begin{array}{cc}1&t\\0&1\end{array}}\right).}$$

That is, the flow at time ${\displaystyle t}$ starting from a vector represented by ${\displaystyle g\in \mathrm {PSL} _{2}(\mathbb {R} )}$ is equal to ${\displaystyle gu_{t}}$.

If ${\displaystyle S}$ is a

hyperbolic surface

its unit tangent bundle also supports a horocycle flow. If ${\displaystyle S}$ is uniformised as ${\displaystyle S=\Gamma \backslash \mathbb {H} ^{2}}$ the unit tangent bundle is identified with ${\displaystyle \Gamma \backslash \mathrm {PSL} _{2}(\mathbb {R} )}$ and the flow starting at ${\displaystyle \Gamma g}$ is given by ${\displaystyle t\mapsto \Gamma gu_{t}}$. When ${\displaystyle S}$ is compact, or more generally when ${\displaystyle \Gamma }$ is a

See also

• Horosphere
• Hypercycle (geometry)

References

• John Wiley & Sons
  .
• Four Pillars of Geometry p. 198