{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\n*Incidence geometry* is the study of [[incidence structures|FGKNAFNHA3]] $(P, L, I)$ and of their properties that are preserved by isomorphisms.\nAn isomorphism of incidence structures is a pair of bijections, on points and on lines, that preserves incidence in both directions.\nThe structures studied are usually required to satisfy incidence axioms, such as:\n\n1. any two distinct points lie on at most one common line (a *partial linear space*);\n2. any two distinct points lie on exactly one common line (a *linear space*).\n\n## Remarks\n\nExample: the statement \"two distinct lines of a linear space meet in at most one point\" belongs to incidence geometry.\nCheck: if two distinct lines had two common points, these points would lie on two lines, against clause 2.\n\nNon-example: the statement \"the angles of a triangle add up to $\\pi$\" is not a statement of incidence geometry, since angles are not defined from points, lines and incidence alone.\n\nMain classes are [[affine planes|EC7XT2ZM5S]], [[projective planes|DANFNN8C4Y]], projective spaces, polar spaces and generalized polygons.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CSXT0BG2CM",
    "kind": "definition",
    "msc": [
      "51A05",
      "51E30"
    ],
    "name": "Incidence geometry",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Incidence_geometry"
      }
    ],
    "tag": "H5EWB180S6",
    "uses": [
      "FGKNAFNHA3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/H5EWB180S6/",
  "links": [
    {
      "field": "defines",
      "target": "CSXT0BG2CM"
    },
    {
      "field": "uses",
      "target": "FGKNAFNHA3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/H5EWB180S6.md",
  "msc": [
    "51A05",
    "51E30"
  ],
  "name": "Incidence geometry",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H5E/H5EWB180S6.md",
  "status": null,
  "tag": "H5EWB180S6",
  "uri": "m2://H5EWB180S6",
  "url": "https://tamnd.github.io/math2/wiki/incidence-geometry-definition/",
  "writing": {
    "tag": "H5EWB180S6",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}