Horocycle
In hyperbolic geometry, a horocycle ( — border + circle, sometimes called an oricycle, oricircle, or limit circle) is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere (or "orisphere"). The centre of a horocycle is the ideal point where all normal geodesics asymptotically converge. Two horocycles who have the same centre are concentric. Although it appears as if two concentric horocycles cannot have the same length or curvature, in fact any two horocycles are congruent. A horocycle can also be described as the limit of  the circles that share a tangent in a given point, as their radii go towards infinity. In Euclidean geometry, such a "circle of infinite radius" would be a straight line, but in hyperbolic geometry it is a horocycle (a curve). From the convex side the horocycle is approximated by hypercycles whose distances from their axis go towards infinity. Properties. Standardized Gaussian curvature. When the hyperbolic plane has the standardized Gaussian curvature "K" of −1: Representations in models of hyperbolic geometry. Poincaré disk model. In the Poincaré disk model of the hyperbolic plane, horocycles are represented by circles tangent to the  boundary circle, the centre of the horocycle is the ideal point where the horocycle touches the boundary circle. The compass and straightedge construction of the two horocycles through two points is the same construction of the CPP construction for the Special cases of Apollonius' problem where both points are inside the circle. Poincaré half-plane model. In the Poincaré half-plane model, horocycles are represented by circles tangent to the boundary line, in which case their centre is the ideal point where the circle touches the boundary line. When the centre of the horocycle is the ideal point at formula_2  then the horocycle is a line parallel to the boundary line. The compass and straightedge construction in the first case is the same construction as the LPP construction for the Special cases of Apollonius' problem. Hyperboloid model. In the hyperboloid model they are represented by intersections of the hyperboloid with planes whose normal lies in the asymptotic cone. Metric. If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of geodesic curvature 1 at every point.