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  "body": "\n## Example\n\nThe *Beltrami–Klein model* of the [[hyperbolic plane|GN4N94GQ0Z]] is the open unit disc $D = \\set{x \\in \\R^2 : \\abs{x} \u003c 1}$ with the following distance.\nFor distinct points $p, q \\in D$, the line through $p$ and $q$ meets the unit circle in two points $a$ and $b$.\nLabel them so that the order along the line is $a, p, q, b$.\nThen\n$$d(p, q) = \\frac{1}{2} \\ln \\frac{\\abs{aq} \\, \\abs{pb}}{\\abs{ap} \\, \\abs{qb}},$$\nwhere $\\abs{xy}$ is the Euclidean distance, and $d(p, p) = 0$.\nThis is the [[Cayley–Klein metric|J7ZE3EGRYX]] whose absolute is the unit circle, and it makes $D$ a [[metric space|0VKD1TP7F3]].\nThe lines of the model are the open chords of the disc.\n\nCheck on a diameter: take $p = (0, 0)$ and $q = (r, 0)$ with $0 \u003c r \u003c 1$.\nThen $a = (-1, 0)$ and $b = (1, 0)$, so $\\abs{aq} = 1 + r$, $\\abs{pb} = 1$, $\\abs{ap} = 1$ and $\\abs{qb} = 1 - r$.\nThe formula gives $d(p, q) = \\frac{1}{2} \\ln \\frac{1 + r}{1 - r} = \\operatorname{artanh} r$, which tends to $\\infty$ as $r \\to 1$.\nSo the boundary circle lies at infinite distance, and the disc is a whole infinite plane.\n\n## Remarks\n\nThrough a point $p$ not on a chord $\\ell$ there pass infinitely many chords that do not meet $\\ell$ inside the disc, so the parallel postulate fails.\nThe isometries of the model are the projective transformations of the plane that map the disc onto itself.\nThe model is not conformal: Euclidean angles between chords are in general not the hyperbolic angles.\nThe map $u \\mapsto 2u / (1 + \\abs{u}^2)$ is an isometry from the [[Poincaré disk model|85MM2EA0TK]] onto the Beltrami–Klein model.\nIndeed, a point at Poincaré distance $t$ from the centre has Euclidean radius $\\tanh(t/2)$ in the Poincaré disc and $\\tanh t$ in the Klein disc, and $\\tanh t = 2 \\tanh(t/2) / (1 + \\tanh^2(t/2))$.\n\n## Explanations\n\n### L1 human\n\nThe Beltrami–Klein model draws the hyperbolic plane inside a round disc, with straight chords as its lines.\nBecause lines look straight, problems about lines and their crossings become problems of ordinary straight-line drawing.\nThe price is that distances and angles are distorted: lengths blow up near the rim, and right angles do not look right.\nFor example, the segment from the centre to the point at Euclidean radius $0.9$ has hyperbolic length $\\operatorname{artanh} 0.9 \\approx 1.47$.\n",
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