{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $z_1, z_2, z_3, z_4$ be four distinct complex numbers.\nTheir cross-ratio is\n$$(z_1, z_2; z_3, z_4) = \\frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}.$$\nSuppose that one of the four points is the point $\\infty$ of the [[Riemann sphere|0CDMXGPSES]] and the other three are distinct complex numbers. Then the cross-ratio is the same formula without the two factors that contain $\\infty$.\n\n## Remarks\n\nOther books use other orders of the four points in the formula, so one must check the convention.\nFor each [[Möbius transformation|0B346DDAM5]] $f$, the cross-ratio $(f(z_1), f(z_2); f(z_3), f(z_4))$ is equal to $(z_1, z_2; z_3, z_4)$.\nWith this convention, the map $z \\mapsto (z, z_2; z_3, z_4)$ is the Möbius transformation that sends $z_2$, $z_3$, $z_4$ to $1$, $0$, $\\infty$.\nThe cross-ratio is a real number exactly when the four points are on one circle or on one line.\n\n## Explanations\n\n### L1 human\n\nThe cross-ratio is one number made from four points.\nMöbius transformations change distances and angles of figures in complicated ways, but they do not change this number.\nSo the cross-ratio helps to find a Möbius map that sends given points to other given points.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "09ASMRMD67",
    "kind": "definition",
    "msc": [
      "30C35"
    ],
    "name": "Cross-ratio",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Cross-ratio"
      },
      {
        "locator": "Transformational approach",
        "url": "https://en.wikipedia.org/wiki/Cross-ratio"
      }
    ],
    "tag": "05MW7MPRBS",
    "uses": [
      "0CDMXGPSES"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/05MW7MPRBS/",
  "links": [
    {
      "field": "defines",
      "target": "09ASMRMD67"
    },
    {
      "field": "uses",
      "target": "0CDMXGPSES"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05MW7MPRBS.md",
  "msc": [
    "30C35"
  ],
  "name": "Cross-ratio",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05M/05MW7MPRBS.md",
  "status": null,
  "tag": "05MW7MPRBS",
  "uri": "m2://05MW7MPRBS",
  "url": "https://math2.tamnd.com/wiki/cross-ratio-definition/",
  "writing": {
    "tag": "05MW7MPRBS",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}