horocycle

A horocycle is a geometric object that arises in the study of hyperbolic geometry. It is defined as a set of points in hyperbolic space that are equidistant from a given geodesic. Think of it as a "hyperbolic circle" that is centered on a line rather than a point.

More formally, in the upper half-plane model of hyperbolic geometry, a horocycle can be represented in a few ways. One common representation is a set of points $(x, y)$ with $y > 0$ such that $x^2 + y^2 = r^2$ for some fixed $r > 0$. In this case, the horocycle is a semicircle with its center at the origin $(0,0)$. However, the origin is not part of the hyperbolic plane. The geodesic that the horocycle is equidistant from is the vertical line $x=0$, which is the y-axis. Another representation is a set of points $(x, y)$ with $y > 0$ such that $y = c$ for some fixed $c > 0$. These are horizontal lines in the upper half-plane, and they are equidistant from the "point at infinity" on the x-axis.

Horocycles play a role in various areas of mathematics, including the study of Kleinian groups and dynamical systems. Their unique curvature and relationship to geodesics make them important structures for understanding the geometry of hyperbolic spaces. They are not to be confused with ordinary circles, as their geometric properties differ significantly due to the non-Euclidean nature of hyperbolic space.