Horocycle

In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics (normals) through a point on a horocycle are limiting parallel, and all converge asymptotically to a single ideal point called the centre of the horocycle.

In some models of hyperbolic geometry, it looks like the two "ends" of a horocycle get closer and closer to each other and closer to its centre, but this is not true; the two "ends" of a horocycle get further and further away from each other and stay at an infinite distance off its centre.

A horosphere is the 3-dimensional version of a horocycle.

In Euclidean space, all curves of constant curvature are either straight lines (geodesics) or circles, but in a hyperbolic space of sectional curvature

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the curves of constant curvature come in four types: geodesics with curvature

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hypercycles with curvature

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horocycles with curvature

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and circles with curvature

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

Properties

Properties similar to those of Euclidean circles

Horocycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:

No three points of a horocycle are on a line, circle or hypercycle.

Three points that are not on a line, circle or hypercycle are on a horocycle

A horocycle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry.