{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "PGCESD48XJ"
    },
    {
      "field": "uses",
      "target": "TZHD2WKNSD"
    }
  ],
  "body": "\n## Definition\n\nLet $U$ and $V$ be [[open sets|02BH3MCBQB]] in $\\C$ and let $K \\ge 1$.\nWrite $f_z = \\frac{1}{2}(f_x - i f_y)$ and $f_{\\bar z} = \\frac{1}{2}(f_x + i f_y)$ for the [[Wirtinger derivatives|WMXDVF36ZT]] of a function $f$, where $f_x$ and $f_y$ are its partial derivatives.\nA map $f \\colon U \\to V$ is *$K$-quasiconformal* if:\n\n1. $f$ is an orientation-preserving [[homeomorphism|0QT00PB5G2]] of $U$ onto $V$.\n2. $f$ has locally square-integrable weak first partial derivatives on $U$, that is, $f$ lies in the Sobolev space $W^{1,2}_{\\mathrm{loc}}(U)$.\n3. $\\abs{f_{\\bar z}} \\le k \\abs{f_z}$ almost everywhere on $U$, where $k = (K - 1)/(K + 1)$.\n\nA map is *quasiconformal* if it is $K$-quasiconformal for some $K \\ge 1$.\n\n## Remarks\n\nExample: for $0 \\le k \u003c 1$, the real-linear map $f(z) = z + k \\bar z$ has $f_z = 1$ and $f_{\\bar z} = k$.\nIts Jacobian determinant is $\\abs{f_z}^2 - \\abs{f_{\\bar z}}^2 = 1 - k^2 \u003e 0$, so it is an orientation-preserving homeomorphism of $\\C$, and it is $K$-quasiconformal with $K = (1 + k)/(1 - k)$.\nIt maps the unit circle onto an ellipse with axis ratio $(1 + k)/(1 - k)$.\nNon-example: complex conjugation $f(z) = \\bar z$ is a homeomorphism, but $f_z = 0$ and $f_{\\bar z} = 1$, so clause 3 fails for every $K$; it reverses orientation.\nThe quotient $\\mu_f = f_{\\bar z} / f_z$ is the complex dilatation of $f$, and $K_f = (1 + \\norm{\\mu_f}_\\infty)/(1 - \\norm{\\mu_f}_\\infty)$ is its maximal dilatation.\nA map is $1$-quasiconformal if and only if it is conformal.\nAn equivalent geometric definition uses the moduli of quadrilaterals or annuli: $f$ is $K$-quasiconformal when it distorts each [[conformal modulus|2FQAKBVBX5]] by a factor of at most $K$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "SB8KG46NHP",
    "kind": "definition",
    "msc": [
      "30C62",
      "30C65"
    ],
    "name": "Quasiconformal mapping",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quasiconformal_mapping"
      }
    ],
    "tag": "A9J2KK4CEJ",
    "uses": [
      "0QT00PB5G2",
      "WMXDVF36ZT",
      "02BH3MCBQB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/A9J2KK4CEJ/",
  "links": [
    {
      "field": "defines",
      "target": "SB8KG46NHP"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    },
    {
      "field": "uses",
      "target": "WMXDVF36ZT"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/A9J2KK4CEJ.md",
  "msc": [
    "30C62",
    "30C65"
  ],
  "name": "Quasiconformal mapping",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A9J/A9J2KK4CEJ.md",
  "status": null,
  "tag": "A9J2KK4CEJ",
  "uri": "m2://A9J2KK4CEJ",
  "url": "https://math2.tamnd.com/wiki/quasiconformal-mapping-definition/",
  "writing": {
    "tag": "A9J2KK4CEJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}