page_title: Horocycle
hierarchical_section_title: Horocycle
context_page_description: In hyperbolic geometry, a horocycle is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional example of a horosphere.
  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.
  From the convex side the horocycle is approximated by hypercycles whose distances from their axis go towards infinity.
context_section_description: 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.
caption: A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central ideal point.