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Concept
Horocycle
Summary
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 (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.
Through every pair of points there are 2 horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the segment between them.
No three points of a horocycle are on a line, circle or hypercycle.
A straight line, circle, hypercycle, or other horocycle cuts a horocycle in at most two points.
The perpendicular bisector of a chord of a horocycle is a normal of the horocycle and it bisects the arc subtended by the chord.
The length of an arc of a horocycle between two points is:
longer than the length of the line segment between those two points,
longer than the length of the arc of a hypercycle between those two points and
shorter than the length of any circle arc between those two points.
The distance from a horocycle to its center is infinite, and while in some models of hyperbolic geometry it looks like the two "ends" of a horocycle get closer and closer together and closer to its center, this is not true; the two "ends" of a horocycle get further and further away from each other.
A regular apeirogon is circumscribed by either a horocycle or a hypercycle.
Official source
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