Horocycle

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A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central ideal point.

In hyperbolic geometry, a horocycle (Greek: ὅριον + κύκλος — border + circle, sometimes called an oricycle , oricircle, or limit circle ) is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction . It is the two-dimensional example of a horosphere (or orisphere).

The centre of a horocycle is the ideal point where all normal geodesics asymptotically converge. Two horocycles who have the same centre are concentric. While it looks that two concentric horocycles cannot have the same length or curvature, in fact any two horocycles are congruent.

A horocycle can also be described as the limit of the circles that share a tangent in a given point, as their radii go towards infinity. In Euclidean geometry, such a "circle of infinite radius" would be a straight line, but in hyperbolic geometry it is a horocycle (a curve) .

From the convex side the horocycle is approximated by hypercycles whose distances from their axis go towards infinity.

Properties[edit]

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Hyperbolic apeirogon example.png
  • 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.
  • 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 the horocycle and it bisects the arc subtended by the chord.
  • 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 an hypercycle between those two points and
shorter than the length of any circle arc between those two point.
  • A regular apeirogon is circumscribed by a horocycle.
  • 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]

Standardized Gaussian curvature[edit]

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:
 s = 2 \sinh \left( \frac{1}{2} d \right) where d is the distance between the two points. [2]
  • 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.[3]
  • The ratio of the arc lengths between two radii of two horocycles where the horocycles are a distance 1 apart is e : 1.[4]

Representations in models of hyperbolic geometry[edit]

The order-3 apeirogonal tiling, {∞,3}, fills the hyperbolic plane with apeirogons whose vertices exist along horocyclic paths.


Poincaré disk model[edit]

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.

Poincaré half-plane model[edit]

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 is the point where the circle touches the boundary line.

When the centre of the horocycle is the ideal point at  y = \infty 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.

Hyperboloid model[edit]

In the hyperboloid model they are represented by intersections of the hyperboloid with planes whose normal lies in the asymptotic cone.

Metric[edit]

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

See also[edit]

Circles seen in a Apollonian gasket that are tangent to the external circle can be considered horocycles in a Poincare disk model

References[edit]

  1. ^ Sossinsky, A.B. (2012). Geometries. Providence, R.I.: American Mathematical Society. pp. 141–2. ISBN 9780821875711. 
  2. ^ Smogorzhevsky (1976). Lobachevskian Geometry. Moscow: Mir. p. 65. 
  3. ^ Sommerville, D.M.Y. (2005). The elements of non-Euclidean geometry (Unabr. and unaltered republ. ed.). Mineola, N.Y.: Dover Publications. p. 58. ISBN 0-486-44222-5. 
  4. ^ Sommerville, D.M.Y. (2005). The elements of non-Euclidean geometry (Unabr. and unaltered republ. ed.). Mineola, N.Y.: Dover Publications. p. 58. ISBN 0-486-44222-5.