{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $S$ and $S'$ be [[projective geometries|N32Z68F0D3]], with point sets $P$ and $P'$.\nA *collineation* from $S$ to $S'$ is a bijection $f \\colon P \\to P'$ with the following property.\nFor all points $a, b, c$ of $P$, the points $a, b, c$ are [[collinear|0AFNNTRTJE]] exactly when $f(a), f(b), f(c)$ are collinear.\nA collineation from $S$ to itself is a collineation of $S$.\n\n## Remarks\n\nEquivalently, $f$ is a bijection on points such that the image of every line is a line and every line of $S'$ is such an image.\nThe same definition applies to affine spaces and to other incidence structures whose lines are sets of points.\n\nExample: every [[projective transformation|P1MKTF7MWQ]] $[v] \\mapsto [Tv]$ of $\\mathbb{P}(V)$ is a collineation.\nCheck: the line through $[u]$ and $[w]$ is $\\mathbb{P}(W)$ for the plane $W$ spanned by $u, w$.\nThe map sends it onto $\\mathbb{P}(T W)$, again a line, and the inverse map comes from $T^{-1}$.\n\nNon-example: in the Fano plane $\\mathbb{P}^2(\\F_2)$, whose lines are the sets $\\set{a, b, a + b}$ of nonzero vectors of $\\F_2^3$, swap $e_1$ and $e_2$ and fix all other points.\nThe line $\\set{e_1, e_3, e_1 + e_3}$ goes to $\\set{e_2, e_3, e_1 + e_3}$, which is not a line, because $e_2 + e_3 \\ne e_1 + e_3$.\n\nThe collineations of $S$ form a group under composition, the collineation group of $S$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "V7PRPN24CV",
    "kind": "definition",
    "msc": [
      "51A10",
      "51N15"
    ],
    "name": "Collineation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Collineation"
      }
    ],
    "tag": "4695YA7F87",
    "uses": [
      "N32Z68F0D3",
      "0AFNNTRTJE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/4695YA7F87/",
  "links": [
    {
      "field": "defines",
      "target": "V7PRPN24CV"
    },
    {
      "field": "uses",
      "target": "N32Z68F0D3"
    },
    {
      "field": "uses",
      "target": "0AFNNTRTJE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4695YA7F87.md",
  "msc": [
    "51A10",
    "51N15"
  ],
  "name": "Collineation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/469/4695YA7F87.md",
  "status": null,
  "tag": "4695YA7F87",
  "uri": "m2://4695YA7F87",
  "url": "https://tamnd.github.io/math2/wiki/collineation-definition/",
  "writing": {
    "tag": "4695YA7F87",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}