{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P$ be a finite set of points in $\\R^2$, not all [[collinear|0AFNNTRTJE]], and let $\\operatorname{conv}(P)$ be its [[convex hull|0FQSDC2EQ2]].\nA *triangulation* of $P$ is a finite set $T$ of closed non-degenerate [[triangles|0CTNFGYMM2]] such that:\n\n1. the vertices of every triangle of $T$ belong to $P$;\n2. any two distinct triangles of $T$ are either disjoint or meet in one common vertex or in one common edge;\n3. the union of the triangles of $T$ is $\\operatorname{conv}(P)$;\n4. every point of $P$ is a vertex of some triangle of $T$.\n\n## Remarks\n\nExample: let $P = \\set{(0,0), (1,0), (1,1), (0,1)}$.\nThe two triangles with vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ form a triangulation: they meet along the diagonal, and their union is the unit square.\n\nNon-example: add the centre $(1/2, 1/2)$ to $P$ and keep the same two triangles.\nClauses 1 to 3 still hold, but the centre is not a vertex of either triangle, so clause 4 fails.\n\nIf $P$ has $n$ points, of which $h$ lie on the boundary of $\\operatorname{conv}(P)$, then every triangulation of $P$ has $2n - h - 2$ triangles and $3n - h - 3$ edges.\nSome authors drop clause 4 and allow points of $P$ that are not used as vertices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "V0B6C6TMWD",
    "kind": "definition",
    "msc": [
      "52C99",
      "68U05",
      "52B70"
    ],
    "name": "Triangulation of a point set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Point-set_triangulation"
      }
    ],
    "tag": "6MFS8FEYHD",
    "uses": [
      "0FQSDC2EQ2",
      "0CTNFGYMM2",
      "0AFNNTRTJE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6MFS8FEYHD/",
  "links": [
    {
      "field": "defines",
      "target": "V0B6C6TMWD"
    },
    {
      "field": "uses",
      "target": "0FQSDC2EQ2"
    },
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    },
    {
      "field": "uses",
      "target": "0AFNNTRTJE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6MFS8FEYHD.md",
  "msc": [
    "52C99",
    "68U05",
    "52B70"
  ],
  "name": "Triangulation of a point set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6MF/6MFS8FEYHD.md",
  "status": null,
  "tag": "6MFS8FEYHD",
  "uri": "m2://6MFS8FEYHD",
  "url": "https://tamnd.github.io/math2/wiki/triangulation-of-a-point-set-definition/",
  "writing": {
    "tag": "6MFS8FEYHD",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}