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In hyperbolic geometry, a horocycle ((ギリシア語: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. == Properties == ((ps still in draft)) * 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 an hypercycle between those two points and :: shorter than the length of any circle arc between those two point. * A regular apeirogon is circumscribed by a horocycle. *If ''C'' is the centre of a horocycle and ''A'' and ''B'' are points on the horocycle then the angles CAB and CBA are equal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Horocycle」の詳細全文を読む スポンサード リンク
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