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  "body": "\n## Definition\n\nLet $\\hat{\\C} = \\C \\cup \\set{\\infty}$ be the extended plane, the [[Riemann sphere|0J78KBPFZB]].\nA *generalized circle* of $\\hat{\\C}$ is either a Euclidean [[circle|060W5CBZP6]] or a set $g \\cup \\set{\\infty}$ with $g$ a line.\nExtend each [[inversion in a circle|YXST7NGMTC]] to $\\hat{\\C}$ by swapping its center with $\\infty$, and each [[reflection|06BRDF3F0E]] in a line by fixing $\\infty$.\nThe *inversive group* is the group of bijections of $\\hat{\\C}$ generated by these maps.\n*Inversive geometry* is the study of the properties of subsets of $\\hat{\\C}$ that are invariant under the inversive group.\n\n## Remarks\n\nIn the language of the [[Erlangen program|9QW65PGEPW]], inversive geometry is the geometry of $\\hat{\\C}$ with the inversive group as its group of transformations.\nThe elements of the inversive group that preserve orientation are the [[Möbius transformations|0SG74WBFA2]] $z \\mapsto (az + b)/(cz + d)$ with $ad - bc \\ne 0$.\nThe others are the maps $z \\mapsto (a\\bar{z} + b)/(c\\bar{z} + d)$ with $ad - bc \\ne 0$.\nEvery element maps generalized circles to generalized circles and preserves the size of angles between curves.\n\nExample: the property \"these four points lie on a common generalized circle\" is inversive, since the inversive group maps generalized circles to generalized circles.\n\nNon-example: the property \"this circle has radius $1$\" is not inversive; the inversion $z \\mapsto 4 / \\bar{z}$ maps the circle $\\abs{z} = 1$ to the circle $\\abs{z} = 4$.\n\nThe same construction in $\\R^n \\cup \\set{\\infty}$, with inversions in spheres and reflections in hyperplanes, gives inversive geometry in dimension $n$.\n",
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