In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature which converges asymptotically in both directions to a single ideal point, called the centre of the horocycle. The perpendicular geodesics through every point on a horocycle are limiting parallels, and also all converge asymptotically to the centre. It is the two-dimensional case of a horosphere.
In Euclidean space, all curves of constant curvature are either straight lines (geodesics) or circles, but in a hyperbolic space of sectional curvature the curves of constant curvature come in four types: geodesics with curvature hypercycles with curvature horocycles with curvature and circles with curvature
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 concentric . As for conceptric circles, any geodesic perpendicular to a horocycle is also perpendicular to every concentric horocycle.
When the hyperbolic plane has the standardized Gaussian curvature K of −1:
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 compass and straightedge construction of the two horocycles through two points is the same construction of the CPP construction for the Special cases of Apollonius' problem where both points are inside the circle.
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 then the horocycle is a line parallel to the boundary line.
The compass and straightedge construction in the first case is the same construction as the LPP construction for the Special cases of Apollonius' problem.
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.)
If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of geodesic curvature 1 at every point.
Every horocycle is the orbit of a unipotent subgroup of PSL(2,R) in the hyperbolic plane. Moreover, the displacement at unit speed along the horocycle tangent to a given unit tangent vector induces a flow on the unit tangent bundle 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 , where:
That is, the flow at time starting from a vector represented by is equal to .
If is a hyperbolic surface its unit tangent bundle also supports a horocycle flow. If is uniformised as the unit tangent bundle is identified with and the flow starting at is given by . When is compact, or more generally when is a lattice, this flow is ergodic (with respect to the normalised Liouville measure). Moreover, in this setting Ratner's theorems describe very precisely the possible closures for its orbits. [7]
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