{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nAn *affine plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:\n\n1. Any two distinct points lie on exactly one common line.\n2. For every line $\\ell$ and every point $p$ not on $\\ell$, exactly one line through $p$ has no point in common with $\\ell$.\n3. There are three points that are not [[collinear|0AFNNTRTJE]].\n\nTwo lines are *parallel* if they are equal or have no common point.\n\n## Remarks\n\nExample: for a field $k$, the points of $k^2$ and the lines $\\set{(x, y) : ax + by = c}$ with $(a, b) \\ne (0, 0)$ form an affine plane.\nCheck: two distinct points determine one such equation up to a factor.\nThe parallels to $ax + by = c$ are the lines $ax + by = c'$, and exactly one of them passes through a given point.\nFinally, $(0,0)$, $(1,0)$, $(0,1)$ are not collinear.\nFor $k = \\F_2$ this is the smallest affine plane, with $4$ points and $6$ lines.\n\nNon-example: the real [[projective plane|DANFNN8C4Y]] satisfies clauses 1 and 3 but not clause 2, because any two of its lines meet.\n\nClause 2 is the parallel axiom in the form of [[Playfair's axiom|PR7Q3QMEHZ]].\nParallelism is an equivalence relation on the lines of an affine plane.\nIn a finite affine plane all lines have the same number $n$ of points, called the order, and there are $n^2$ points.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "EC7XT2ZM5S",
    "kind": "definition",
    "msc": [
      "51A15",
      "51E15"
    ],
    "name": "Affine plane",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)"
      }
    ],
    "tag": "XD8ZB63H78",
    "uses": [
      "FGKNAFNHA3",
      "0AFNNTRTJE",
      "PR7Q3QMEHZ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/XD8ZB63H78/",
  "links": [
    {
      "field": "defines",
      "target": "EC7XT2ZM5S"
    },
    {
      "field": "uses",
      "target": "FGKNAFNHA3"
    },
    {
      "field": "uses",
      "target": "0AFNNTRTJE"
    },
    {
      "field": "uses",
      "target": "PR7Q3QMEHZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XD8ZB63H78.md",
  "msc": [
    "51A15",
    "51E15"
  ],
  "name": "Affine plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XD8/XD8ZB63H78.md",
  "status": null,
  "tag": "XD8ZB63H78",
  "uri": "m2://XD8ZB63H78",
  "url": "https://tamnd.github.io/math2/wiki/affine-plane-definition/",
  "writing": {
    "tag": "XD8ZB63H78",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}