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  "body": "\n## Definition\n\nLet $\\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] of constant curvature $-1$.\nLet $\\ell$ be a line and $P$ a point not on $\\ell$.\nLet $Q$ be the foot of the perpendicular from $P$ to $\\ell$, and let $a = d(P, Q) \u003e 0$.\nA ray from $P$ meets $\\ell$ when its [[angle|0GJAJC6TNZ]] with the ray $PQ$ is small.\nFix a side of the line $PQ$, and for $\\theta \\in [0, \\pi/2]$ let $r_\\theta$ be the ray from $P$ on that side at angle $\\theta$ with $PQ$.\nThe *angle of parallelism* $\\Pi(a)$ is the supremum of the angles $\\theta$ such that $r_\\theta$ meets $\\ell$.\nThe ray at angle exactly $\\Pi(a)$ does not meet $\\ell$; it is a *limiting parallel* to $\\ell$ through $P$.\n\n## Remarks\n\nThe value does not depend on the side of $PQ$, nor on $\\ell$ and $P$, only on $a$, and Lobachevsky's formula gives\n$$\\tan \\frac{\\Pi(a)}{2} = e^{-a}, \\qquad \\text{equivalently} \\qquad \\cos \\Pi(a) = \\tanh a, \\quad \\sin \\Pi(a) = \\frac{1}{\\cosh a}.$$\nExample: in the [[Beltrami–Klein model|1WN9B18H86]], take $P$ the centre and $\\ell$ the chord $x = r$ with $0 \u003c r \u003c 1$.\nLines through the centre are diameters, so the foot is $Q = (r, 0)$ and $a = \\operatorname{artanh} r$.\nThe limiting parallels are the chords from $P$ to the endpoints $(r, \\pm\\sqrt{1 - r^2})$, which make angle $\\theta$ with $\\cos \\theta = r$ at the centre; angles at the centre are Euclidean in this model.\nSo $\\cos \\Pi(a) = r = \\tanh a$, as the formula says.\nNon-example: in the Euclidean plane every ray from $P$ at angle less than $\\pi/2$ with $PQ$ meets $\\ell$, and the ray at angle $\\pi/2$ is the unique parallel.\nThere the supremum is $\\pi/2$ for every $a$, so the angle carries no information about distance.\nIn curvature $-k^2$, the formula holds with $a$ replaced by $ka$.\n",
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