Horocycle

Main article: Horocycle

Further information: Hyperbolic geometry

Definition

In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a special kind of curve. It has constant curvature.

Properties

All the perpendicular geodesics (normals) through any point on a horocycle are limiting parallel. These geodesics all come close to meeting at a single distant point, called the centre of the horocycle.

Relation to Other Curves

In hyperbolic space, curves of constant curvature can be of different types. Horocycles are one of these types, and they are different from the straight lines and circles found in regular Euclidean space. Any two horocycles are the same size and shape; you can move one to match the other by sliding and turning the hyperbolic plane.

Horocycles can also be thought of as what happens when you take circles that all touch at one point and make their sizes grow larger and larger. As the circles get bigger, they become horocycles. Two horocycles that share the same centre are called concentric, and any line that is perpendicular to one will also be perpendicular to all the others with the same centre.

Properties

Horocycles are special curves in hyperbolic geometry. They have some properties like circles in regular geometry. For example, three points that don’t lie on a straight line can always form a horocycle.

A line drawn perpendicular to a radius at its endpoint on the horocycle will touch the horocycle at just one point, like a tangent for a circle.

All horocycles are the same size no matter where their center is. The area between two lines connecting to the center of a horocycle is finite, but the whole horocycle has an infinite area. The ends of a horocycle may look like they come closer together, but they actually move farther apart as you go out.

Representations in models of hyperbolic geometry

In the Poincaré disk model, horocycles look like circles that just touch the edge of the big circle. The center of a horocycle is where it touches the edge. In real hyperbolic geometry, every point on a horocycle is very far from its center. The distance between points at opposite ends gets larger as the points move farther apart.

In the Poincaré half-plane model, horocycles look like circles that touch the bottom line, with their center at the point where they touch. If the center is far away, the horocycle looks like a straight line parallel to the bottom edge. In the hyperboloid model, horocycles are shown where the hyperboloid meets certain flat surfaces that make parabolas.

Metric

When we change the rules of measuring space to give it a special curved shape, a horocycle becomes a special kind of curve. In this setup, the horocycle bends the same gentle amount at every point along its path. This steady bending helps mathematicians learn about the unique features of horocycles in these special curved spaces.

Horocycle flow

Every horocycle is linked to special changes in hyperbolic geometry. Moving along a horocycle at a steady speed creates what is called a horocycle flow. This flow helps us understand how points move and change on the hyperbolic plane.

When we study surfaces with constant negative curvatures, we can also see horocycle flows. These flows have interesting patterns and follow specific rules, showing how points travel along the surface in a predictable way.