“In hyperbolic geometry, a horocycle, 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.”
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Horocycle is just a horosphere of a specific dimension.
In the upper half-space model of hyperbolic geometry any vertical line is a geodesic. All of those lines converge asymptotically to the unique “point at infinity” of this model. Any horizontal plane is perpendicular to any vertical line, therefore this plane along with “the point at infinity” is a horosphere.
In Poincaré model, for example, horospheres actually look like Euclidean spheres. If you pick a “point on the boundary” of this model, the circular arcs that are perpendicular to the boundary at that point are the geodesics of this model and the horospheres are spheres that are tangent to the boundary at that point.
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