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  "body": "\n## Description\n\nThe *Poincaré disk model* of the hyperbolic plane is the [[unit disc|8G5ABXRK85]] $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ with the Riemannian metric\n$$ds = \\frac{2\\, \\abs{dz}}{1 - \\abs{z}^2}.$$\nThe distance between two points is the infimum of the lengths of piecewise smooth curves joining them, which makes $\\mathbb{D}$ a [[metric space|0VKD1TP7F3]].\nThe distance from $0$ to $z$ is\n$$d(0, z) = \\int_0^{\\abs{z}} \\frac{2\\, dt}{1 - t^2} = \\log\\frac{1 + \\abs{z}}{1 - \\abs{z}},$$\nattained along the radius, so points near the boundary circle are infinitely far from $0$.\nIn general, $d(z, w) = 2 \\operatorname{artanh} \\abs{\\dfrac{z - w}{1 - \\bar w z}}$.\n\n## Remarks\n\nThe metric has constant Gaussian curvature $-1$.\nThe geodesics are the diameters of $\\mathbb{D}$ and the arcs of circles that meet the unit circle at right angles.\nThe orientation-preserving isometries are the [[Möbius transformations|0SG74WBFA2]] $z \\mapsto e^{i\\theta}(z - a)/(1 - \\bar a z)$ with $\\abs{a} \u003c 1$, which are the conformal automorphisms of $\\mathbb{D}$.\nThe model is conformal: hyperbolic angles equal Euclidean angles.\nThe [[Schwarz–Pick theorem|V1QYA9RW73]] says that every holomorphic self-map of $\\mathbb{D}$ does not increase this distance.\nThe Cayley map $z \\mapsto (z - i)/(z + i)$ is an isometry from the Poincaré half-plane model onto the disk model.\n\n## Explanations\n\n### L1 human\n\nThe Poincaré disk is a picture of a geometry where the parallel postulate fails.\nThe whole infinite hyperbolic plane is squeezed into a round disc, and distances grow without bound toward the rim.\nStraight lines of this geometry appear as diameters or as circular arcs that hit the rim at right angles.\nThrough a point off such a line there are infinitely many lines that never meet it.\nM. C. Escher's Circle Limit prints show tilings of this disc by congruent figures that look smaller and smaller toward the edge.\n",
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