{
  "aliases": [
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  "body": "\n## Statement\n\nA generalized circle in the [[Riemann sphere|0CDMXGPSES]] $\\widehat{\\mathbb{C}}$ is either a circle in $\\mathbb{C}$, or a set $L \\cup \\{\\infty\\}$ where $L$ is a line in $\\mathbb{C}$.\nLet $f$ be a [[Möbius transformation|0B346DDAM5]].\nThen for each generalized circle $C$, the image $f(C)$ is a generalized circle.\n\n## Remarks\n\nA circle can go to a line: the map $z \\mapsto \\frac{1}{z}$ sends the circle $|z - 1| = 1$, which contains $0$, to the line $\\operatorname{Re} w = \\frac{1}{2}$ together with $\\infty$.\nThe image of a generalized circle is a line together with $\\infty$ exactly when the generalized circle contains the point that $f$ sends to $\\infty$.\nThe proof checks the result for $z \\mapsto z + b$, $z \\mapsto az$ and $z \\mapsto \\frac{1}{z}$, because each Möbius transformation is a composition of such maps.\n\n## Explanations\n\n### L1 human\n\nFor a Möbius map, a line is only a circle through the point $\\infty$.\nSo the theorem says that circles go to circles, if we count lines as circles.\nFor example, $z \\mapsto 2z + 3$ sends each circle to a circle with twice the radius.\n",
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        "locator": "Preservation of angles and generalized circles",
        "url": "https://en.wikipedia.org/wiki/M%C3%B6bius_transformation"
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  "name": "Möbius transformations map circles and lines to circles and lines",
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