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  "body": "\n## Example\n\nThe *Poincaré half-plane model* is the [[upper half-plane|X6XGGH5FPW]] $\\mathbb{H} = \\set{(x, y) \\in \\R^2 : y \u003e 0}$ with the [[Riemannian metric|Q61VFHCH2J]]\n$$ds^2 = \\frac{dx^2 + dy^2}{y^2}.$$\nIts [[geodesics|NSYMQMHPDN]], the lines of the geometry, are the vertical half-lines and the semicircles centred on the $x$-axis.\nThe distance between $(x_1, y_1)$ and $(x_2, y_2)$ is given by\n$$\\cosh d = 1 + \\frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{2 y_1 y_2}.$$\n\nCheck of the distance on the $y$-axis: the length of the segment from $(0, 1)$ to $(0, e)$ is $\\int_1^e dy/y = 1$.\nThe formula gives $\\cosh d = 1 + (e - 1)^2/(2e) = (e^2 + 1)/(2e) = \\cosh 1$, so $d = 1$ as well.\n\nCheck of the hyperbolic parallel axiom: take the line $x = 0$ and the point $(1, 1)$.\nThe vertical line $x = 1$ does not meet it.\nThe semicircle centred at $(2, 0)$ through $(1, 1)$ has radius $\\sqrt 2$, so it stays in $x \\ge 2 - \\sqrt 2 \u003e 0$ and does not meet it either.\nSo $\\mathbb{H}$ is a model of [[hyperbolic geometry|GN4N94GQ0Z]].\n\n## Remarks\n\nThe metric has constant Gaussian curvature $-1$.\nThe orientation-preserving isometries are the [[Möbius transformations|0SG74WBFA2]] $z \\mapsto (az + b)/(cz + d)$ with $a, b, c, d \\in \\R$ and $ad - bc = 1$, written with $z = x + iy$.\nSo the group of these isometries is $\\mathrm{PSL}_2(\\R)$.\nThe model is conformal: hyperbolic angles equal Euclidean angles.\n\n## Explanations\n\n### L1 human\n\nThe half-plane model draws the whole hyperbolic plane inside the region above a horizontal line.\nDistances grow as you approach that line, so the line itself is infinitely far away.\nThe straight lines of the geometry look like vertical rays and half-circles standing on the boundary line.\n",
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