What is "horocycle"

n. (context geometry English) A curve in hyperbolic geometry whose normals all converge asymptotically.

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 example 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. While it looks that 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.