{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P$ be a finite set of points in $\\R^2$, not all collinear.\nFor a triangle $\\Delta$, its *circumdisk* is the open disk bounded by the [[circumcircle|03VZJE6C8C]] of $\\Delta$.\nA [[triangulation|V0B6C6TMWD]] $T$ of $P$ is a *Delaunay triangulation* if no point of $P$ lies in the circumdisk of any triangle of $T$.\n\n## Remarks\n\nExample: let $P = \\set{(0,0), (2,0), (1,2)}$, a single triangle.\nIts only triangulation is the triangle itself, and the open circumdisk contains no point of $P$, since the three points lie on the circle.\n\nNon-example: let $P = \\set{(-2, 0), (2, 0), (0, 1), (0, -1)}$ and take the triangulation by the diagonal from $(-2, 0)$ to $(2, 0)$.\nThe circumcircle of $(-2, 0), (2, 0), (0, 1)$ has centre $(0, -3/2)$ and radius $5/2$, and $(0, -1)$ is at distance $1/2$ from the centre, so it lies inside.\nThe other diagonal, from $(0, 1)$ to $(0, -1)$, gives the Delaunay triangulation.\n\nA Delaunay triangulation of $P$ always exists, and it is unique when no four points of $P$ lie on a common circle.\nAmong all triangulations of $P$, a Delaunay triangulation maximizes the smallest angle of its triangles.\nWhen no four points are concyclic, two points of $P$ are joined by an edge exactly when their cells in the [[Voronoi diagram|QDNEBBVZ84]] share an edge.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "259Y10BYJC",
    "kind": "definition",
    "msc": [
      "52C99",
      "68U05"
    ],
    "name": "Delaunay triangulation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Delaunay_triangulation"
      }
    ],
    "tag": "SGH803FGY4",
    "uses": [
      "V0B6C6TMWD",
      "03VZJE6C8C",
      "QDNEBBVZ84"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SGH803FGY4/",
  "links": [
    {
      "field": "defines",
      "target": "259Y10BYJC"
    },
    {
      "field": "uses",
      "target": "V0B6C6TMWD"
    },
    {
      "field": "uses",
      "target": "03VZJE6C8C"
    },
    {
      "field": "uses",
      "target": "QDNEBBVZ84"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SGH803FGY4.md",
  "msc": [
    "52C99",
    "68U05"
  ],
  "name": "Delaunay triangulation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SGH/SGH803FGY4.md",
  "status": null,
  "tag": "SGH803FGY4",
  "uri": "m2://SGH803FGY4",
  "url": "https://tamnd.github.io/math2/wiki/delaunay-triangulation-definition/",
  "writing": {
    "tag": "SGH803FGY4",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}