{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $O$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] and let $r \u003e 0$.\nThe *inversion in the [[circle|060W5CBZP6]]* with center $O$ and radius $r$ sends each point $P \\ne O$ to the unique point $P'$ such that:\n\n1. $P'$ lies on the ray from $O$ through $P$;\n2. $\\abs{OP} \\cdot \\abs{OP'} = r^2$.\n\n## Remarks\n\nIn coordinates with $O$ at the origin, the inversion is $x \\mapsto r^2 x / \\norm{x}^2$.\nIn complex notation with center $0$, it is $z \\mapsto r^2 / \\bar{z}$.\nAdding a point $\\infty$ and setting $O \\mapsto \\infty$ and $\\infty \\mapsto O$ extends it to a bijection of the [[Riemann sphere|0J78KBPFZB]].\nIt is an [[involution|GP56NBN9RY]]: applying it twice gives the identity.\nThe same formula defines inversion in a sphere of $\\R^n$.\n\nExample: for the circle with center $(0, 0)$ and radius $1$, the point $P = (2, 0)$ goes to $P' = (1/2, 0)$.\nCheck: $P'$ lies on the ray from the origin through $P$, and $\\abs{OP} \\cdot \\abs{OP'} = 2 \\cdot \\frac{1}{2} = 1$.\n\nNon-example: the map $x \\mapsto -r^2 x / \\norm{x}^2$ satisfies clause 2 but sends $P$ to the opposite ray, so clause 1 fails.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "YXST7NGMTC",
    "kind": "definition",
    "msc": [
      "51M05",
      "51B10"
    ],
    "name": "Inversion in a circle",
    "sources": [
      {
        "locator": "Inversion in a circle",
        "url": "https://en.wikipedia.org/wiki/Inversive_geometry"
      }
    ],
    "tag": "G03V3D097M",
    "uses": [
      "0XW7V9ZZ05",
      "060W5CBZP6",
      "GP56NBN9RY",
      "0J78KBPFZB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/G03V3D097M/",
  "links": [
    {
      "field": "defines",
      "target": "YXST7NGMTC"
    },
    {
      "field": "uses",
      "target": "0XW7V9ZZ05"
    },
    {
      "field": "uses",
      "target": "060W5CBZP6"
    },
    {
      "field": "uses",
      "target": "GP56NBN9RY"
    },
    {
      "field": "uses",
      "target": "0J78KBPFZB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/G03V3D097M.md",
  "msc": [
    "51M05",
    "51B10"
  ],
  "name": "Inversion in a circle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G03/G03V3D097M.md",
  "status": null,
  "tag": "G03V3D097M",
  "uri": "m2://G03V3D097M",
  "url": "https://tamnd.github.io/math2/wiki/inversion-in-a-circle-definition/",
  "writing": {
    "tag": "G03V3D097M",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}