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  "body": "\n## Example\n\nLet $\\mathbb{H} = \\{ z \\in \\mathbb{C} : \\operatorname{Im} z \u003e 0 \\}$ be the upper half-plane and $\\mathbb{D} = \\{ w \\in \\mathbb{C} : |w| \u003c 1 \\}$ the open unit disc.\nThe Cayley transform is\n$$W(z) = \\frac{z - i}{z + i}.$$\nIt is a [[Möbius transformation|0B346DDAM5]] with $a = 1$, $b = -i$, $c = 1$, $d = i$, and $ad - bc = 2i \\neq 0$.\nFor $z \\in \\mathbb{C}$, $|W(z)| \u003c 1$ holds exactly when $z$ is nearer to $i$ than to $-i$, that is, when $\\operatorname{Im} z \u003e 0$.\nThe map $W$ is a bijection from $\\mathbb{H}$ onto $\\mathbb{D}$, with inverse $w \\mapsto i \\frac{1 + w}{1 - w}$.\n\n## Remarks\n\nSome values are $W(i) = 0$, $W(0) = -1$, $W(1) = -i$, $W(-1) = i$ and $W(\\infty) = 1$.\nThe map $W$ sends the real line together with $\\infty$ onto the unit circle.\nIt is an explicit case of the [[Riemann mapping theorem|02WHMEG5ZR]] for $\\mathbb{H}$.\nIn hyperbolic geometry, it connects the half-plane model and the disc model.\n\n## Explanations\n\n### L1 human\n\nThe Cayley transform turns the upper half-plane into the unit disc.\nA point of $\\mathbb{H}$ is nearer to $i$ than to $-i$, so the quotient of the two distances is less than $1$.\nIt lets you move a problem on the half-plane to the disc, where it can be easier.\n",
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