hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

whose

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

or perpendicular

geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere (or ''orisphere''). The centre of a horocycle is the

ideal point In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left- limiting parallels to ''l'' through ''P' ...

where all normal geodesics asymptotically converge. Two horocycles who have the same centre are concentric. Although it appears as if two concentric horocycles cannot have the same length or curvature, in fact any two horocycles are

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

. 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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, 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

* 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

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 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 center 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 center, this is not true; the two "ends" of a horocycle get further and further away from each other. * A regular

apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to th ...

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. * The area of a sector of a horocycle (the area between two radii and the horocycle) is finite.

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: :

s = 2 \sinh \left( \frac d \right) = \sqrt

where ''d'' is the distance between the two points, and sinh and cosh are

hyperbolic functions In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...

. * 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. the area enclosed between this horocycle and the radii is 1. * The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is ''e'' : 1.

Representations in models of hyperbolic geometry

Order-3 apeirogonal tiling one cell horocycle

Poincaré disk model

In the Poincaré disk model of the hyperbolic plane, horocycles are represented by circles

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

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

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

y = \infty

then the horocycle is a line parallel to the boundary line. The

in the first case is the same construction as the LPP construction for the Special cases of Apollonius' problem.

Hyperboloid model

In the

hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbolo ...

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

Metric

If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of

geodesic curvature In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's ...

1 at every point.

References

{{Reflist * H. S. M. Coxeter (1961) ''Introduction to Geometry'', §16.6: "Circles, horocycles, and equidistant curves", page 300, 1,

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, ...

.
Four Pillars of Geometry
p. 198 Hyperbolic geometry Curves