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  "body": "\n## Definition\n\nLet $S^2 = \\set{(x_1, x_2, x_3) \\in \\R^3 : x_1^2 + x_2^2 + x_3^2 = 1}$ be the unit sphere and let $N = (0, 0, 1)$ be its north pole.\nIdentify the plane $x_3 = 0$ with $\\C$ by $(x, y, 0) \\mapsto x + iy$.\nThe *stereographic projection* is the map $\\sigma \\colon S^2 \\setminus \\set{N} \\to \\C$ defined as follows.\nFor $P \\in S^2 \\setminus \\set{N}$, the point $\\sigma(P)$ is where the line through $N$ and $P$ meets the plane $x_3 = 0$.\nIn coordinates,\n$$\\sigma(x_1, x_2, x_3) = \\frac{x_1 + i x_2}{1 - x_3}.$$\nSetting $\\sigma(N) = \\infty$ extends $\\sigma$ to a bijection from $S^2$ onto the [[Riemann sphere|0CDMXGPSES]] $\\C \\cup \\set{\\infty}$.\n\n## Remarks\n\nThe inverse map is\n$$\\sigma^{-1}(z) = \\left( \\frac{2 \\operatorname{Re} z}{\\abs{z}^2 + 1}, \\frac{2 \\operatorname{Im} z}{\\abs{z}^2 + 1}, \\frac{\\abs{z}^2 - 1}{\\abs{z}^2 + 1} \\right).$$\nExample: the south pole $(0, 0, -1)$ goes to $0/2 = 0$.\nA point $(x_1, x_2, 0)$ of the equator goes to $x_1 + i x_2$, so the equator goes onto the unit circle.\nNon-example: the projection from the center of the sphere onto a tangent plane is not stereographic.\nIt sends two antipodal points $P$ and $-P$ to the same point, so it is not injective.\nThe map $\\sigma$ is a homeomorphism, it preserves angles, and it sends circles on $S^2$ to circles or lines in $\\C$; the circles through $N$ go to lines.\nSome authors project onto the plane tangent at the south pole, which multiplies $\\sigma$ by $2$.\n",
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