Horocycle

In hyperbolic geometry, a horocycle (Greek: ὅριον + κύκλος — 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.

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When the hyperbolic plane has the standardized Gaussian curvature K of −1:

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.

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