Horocycle

# Horocycle

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In hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, a horocycle ( — border + circle) is a curve whose normals all converge asymptotically. (It is also called an oricycle or oricircle.) It is the two-dimensional example of a horosphere (or orisphere).

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 ordinary euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, such a "circle of infinite radius" would be a straight line, but in hyperbolic geometry it curves. From the convex side the horocycle is approximated by hypercycles
Hypercycle (geometry)
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line.Given a straight line L and a point P not on L,...

whose distances go towards infinity.

In the Poincaré disk model
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...

of the hyperbolic plane, the horocycles are represented by circles tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to the boundary circle. In the Poincaré half-plane model
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....

the horocycles are represented by circles tangent to the boundary line, and lines parallel to the boundary line. In the hyperboloid model
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model , is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are...

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