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  "body": "\n## Summary\n\nTake a line of the [[hyperbolic plane|GN4N94GQ0Z]] and a point at distance $a$ from it.\nThe angle of parallelism $\\Pi(a)$ is the angle between the perpendicular to the line and a limiting parallel through the point.\nIt depends only on the distance $a$, and in curvature $-1$ it is given by $\\tan(\\Pi(a)/2) = e^{-a}$.\nIt decreases from $\\pi/2$ to $0$ as $a$ grows, which measures how far hyperbolic geometry departs from Euclid's [[parallel postulate|2PC3XY0PYZ]].\nThe notation $\\Pi(a)$ goes back to Lobachevsky.\n\n## Explanations\n\n### L1 human\n\nStand at a point $P$ at distance $a$ from a line, and drop the perpendicular to it.\nTurn a ray from the perpendicular toward the line: at first the ray meets the line, and past a certain angle it never does.\nThat critical angle is the angle of parallelism.\nIn Euclidean geometry it is always a right angle, but in hyperbolic geometry it is smaller, and the farther you stand, the smaller it gets.\nFor example, at distance $a = \\ln 3 \\approx 1.1$ one gets $\\tan(\\Pi/2) = 1/3$, so $\\Pi \\approx 37^\\circ$.\n",
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