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  "body": "\n## Summary\n\nA horocycle is a curve of the [[hyperbolic plane|GN4N94GQ0Z]] whose normal geodesics all run to the same point at infinity.\nIt is the limit of circles through a fixed point with a fixed tangent there, as the radius tends to infinity.\nIn the [[Poincaré disk model|85MM2EA0TK]] it is a Euclidean circle inside the disc tangent to the boundary circle, without the point of tangency.\nHorocycles are not geodesics: they have constant geodesic curvature $1$ in curvature $-1$.\n\n## Explanations\n\n### L1 human\n\nIn Euclidean geometry, a circle of huge radius looks almost like a straight line.\nIn hyperbolic geometry this fails: as the radius grows, circles tend to a new kind of curve, the horocycle, which still bends.\nAll the radii of a horocycle, the geodesics that cross it at right angles, head toward one common point at infinity, its centre.\nIn the upper half-plane picture, the horizontal line $y = 1$ is a horocycle: the vertical half-lines cross it at right angles and all go up to the same point at infinity.\n",
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