Horocycle

Summary

A horocycle is a curve of the hyperbolic plane whose normal geodesics all run to the same point at infinity. It is the limit of circles through a fixed point with a fixed tangent there, as the radius tends to infinity. In the Poincaré disk model it is a Euclidean circle inside the disc tangent to the boundary circle, without the point of tangency. Horocycles are not geodesics: they have constant geodesic curvature $1$ in curvature $-1$.