{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "GKV4RVXBRE"
    },
    {
      "field": "uses",
      "target": "GKV4RVXBRE"
    },
    {
      "field": "defines",
      "target": "T7B1WJVT9Z"
    }
  ],
  "body": "\n## Summary\n\nA projective transformation, or homography, is a map between [[projective spaces|ZT9A6JK8QF]] induced by an invertible linear map of the underlying vector spaces.\nIn [[homogeneous coordinates|FS9FRJ17D2]] it is $[x] \\mapsto [Ax]$ for an invertible matrix $A$, defined up to a nonzero factor.\nThe projective transformations of $\\mathbb{P}^n(k)$ form the group $\\mathrm{PGL}(n + 1, k)$.\nThey are the symmetries of [[projective geometry|N32Z68F0D3]], as Euclidean motions are those of Euclidean geometry.\n\n## Explanations\n\n### L1 human\n\nTake a photograph of a flat floor from an angle: a square tile becomes a general quadrilateral.\nThe map from the floor to the photo is a projective transformation.\nIt keeps straight lines straight, but not lengths, angles or parallelism.\nOn the projective line over $\\C$, the Möbius maps $z \\mapsto (az + b)/(cz + d)$ with $ad - bc \\ne 0$ are exactly the projective transformations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "51N15",
      "51A10"
    ],
    "name": "Projective transformation",
    "same_as": {
      "wikidata": "Q2112539",
      "wikipedia": "Homography"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homography"
      }
    ],
    "tag": "P1MKTF7MWQ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/P1MKTF7MWQ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/P1MKTF7MWQ.md",
  "msc": [
    "51N15",
    "51A10"
  ],
  "name": "Projective transformation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P1M/P1MKTF7MWQ.md",
  "status": null,
  "tag": "P1MKTF7MWQ",
  "uri": "m2://P1MKTF7MWQ",
  "url": "https://tamnd.github.io/math2/wiki/projective-transformation/",
  "writing": {
    "tag": "P1MKTF7MWQ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}