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  "body": "\n## Definition\n\nLet $\\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] with distance $d$, and let $\\omega$ be an ideal point, a point at infinity of $\\mathbb{H}^2$.\nLet $\\gamma \\colon [0, \\infty) \\to \\mathbb{H}^2$ be a [[geodesic|NSYMQMHPDN]] ray parametrized by arc length with $\\gamma(t) \\to \\omega$.\nThe *Busemann function* of $\\gamma$ is\n$$b(x) = \\lim_{t \\to \\infty} \\bigl( d(x, \\gamma(t)) - t \\bigr), \\qquad x \\in \\mathbb{H}^2.$$\nA *horocycle centred at* $\\omega$ is a level set $\\set{x \\in \\mathbb{H}^2 : b(x) = c}$ with $c \\in \\R$.\n\n## Remarks\n\nAnother ray to $\\omega$ changes $b$ by an additive constant, so the set of horocycles centred at $\\omega$ depends only on $\\omega$.\nExample: in the [[Poincaré half-plane model|RHKSMPBSQ1]], take $\\omega = \\infty$ and $\\gamma(t) = i e^t$.\nFor $x = u + iy$, the distance formula $\\cosh d(z, w) = 1 + \\abs{z - w}^2 / (2 \\operatorname{Im} z \\operatorname{Im} w)$ gives $d(x, i e^t) = t - \\ln y + o(1)$ as $t \\to \\infty$.\nSo $b(x) = -\\ln y$, and the horocycles centred at $\\infty$ are the horizontal lines $y = \\text{const}$.\nTheir normal geodesics are the vertical half-lines, which all end at $\\infty$.\nHorocycles centred at a real point $\\omega$ are Euclidean circles tangent to the real axis at $\\omega$, without $\\omega$.\nNon-example: the ray $\\set{r e^{i\\theta} : r \u003e 0}$ with $0 \u003c \\theta \u003c \\pi/2$ is not a horocycle.\nIts normal geodesics are the semicircles centred at $0$, and they end at different real points, so no single ideal point $\\omega$ works.\nThis ray is a curve at constant distance from the imaginary axis, called a hypercycle.\n",
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