Horocycle

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, and 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.

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, and they share some interesting properties with circles in regular geometry. For example, just like with circles, three points that don’t lie on a straight line can always form a horocycle. Also, a line drawn perpendicular to a radius at its endpoint on the horocycle will touch the horocycle at just one point, similar to how a tangent works for a circle.

Some other properties include the fact that all horocycles are the same size no matter where their center is. While the area between two lines connecting to the center of a horocycle is finite, the whole horocycle has an infinite area. Even though some drawings might make it look like the ends of a horocycle come closer together, they actually move farther apart as you go out.

Representations in models of hyperbolic geometry

In the Poincaré disk model, horocycles appear as circles that just touch the edge of the circle. The center of a horocycle is the point where it touches the edge. It might seem like points at opposite ends of a horocycle get closer to each other and to the center, but in true hyperbolic geometry, every point on a horocycle is infinitely far from its center. The distance between points at opposite ends actually grows larger as the points move farther apart along the curve.

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 at infinity, the horocycle looks like a straight line parallel to the bottom edge. In the hyperboloid model, horocycles are shown where the hyperboloid meets certain planes that create parabolas.

Metric

When we adjust the measurement rules so that the space has a special curved shape, a horocycle becomes a special curve. In this setup, the horocycle has the same gentle bending at every point along its path. This consistent bending helps mathematicians study the unique properties of horocycles in their special curved worlds.

Horocycle flow

Every horocycle is linked to a special group of transformations 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 curvature, 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.