{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$.\nThe *equi-affine group* of $\\R^{n+1}$ is the group of maps $x \\mapsto Ax + b$ with $A$ in the [[general linear group|0FCHE902D7]] $GL(n+1, \\R)$, $\\det A = 1$ and $b \\in \\R^{n+1}$.\n*Affine differential geometry* is the study of the properties of smooth curves and hypersurfaces in $\\R^{n+1}$ that are invariant under the equi-affine group.\n\n## Remarks\n\nFor a [[regular curve|06JMR2XWXW]] $\\gamma \\colon I \\to \\R^2$ with $\\det(\\gamma', \\gamma'') \u003e 0$, the *affine arc length* is\n$$\\sigma(t) = \\int_{t_0}^{t} \\det\\big(\\gamma'(u), \\gamma''(u)\\big)^{1/3} \\, du.$$\nIt is invariant: if $\\Phi(x) = Ax + b$ with $\\det A = 1$, then $\\det(A\\gamma', A\\gamma'') = \\det(\\gamma', \\gamma'')$.\nFor the parabola $\\gamma(t) = (t, t^2/2)$, one has $\\gamma' = (1, t)$ and $\\gamma'' = (0, 1)$, so the determinant is $1$ and the affine arc length from $0$ is $t$.\n\nEuclidean arc length is not an affine invariant.\nThe shear $(x, y) \\mapsto (x + y, y)$ has determinant $1$, and it maps the segment from $(0,0)$ to $(0,1)$, of length $1$, to the segment from $(0,0)$ to $(1,1)$, of length $\\sqrt{2}$.\n\nFor a nondegenerate hypersurface, the basic invariants are the affine normal vector field and the induced Blaschke metric.\nSome authors use the full affine group, with any invertible $A$; then volume is no longer preserved and the invariants change.\n\n## Explanations\n\n### L1 human\n\nAffine differential geometry replaces the rigid motions of Euclidean geometry by the larger group of area- or volume-preserving affine maps.\nIt asks which features of a curved shape survive all of these maps.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "K5C968HMSN",
    "kind": "definition",
    "msc": [
      "53A15"
    ],
    "name": "Affine differential geometry",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Affine_differential_geometry"
      }
    ],
    "tag": "4ZTTPDXCQT",
    "uses": [
      "0FCHE902D7",
      "06JMR2XWXW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/4ZTTPDXCQT/",
  "links": [
    {
      "field": "defines",
      "target": "K5C968HMSN"
    },
    {
      "field": "uses",
      "target": "0FCHE902D7"
    },
    {
      "field": "uses",
      "target": "06JMR2XWXW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4ZTTPDXCQT.md",
  "msc": [
    "53A15"
  ],
  "name": "Affine differential geometry",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4ZT/4ZTTPDXCQT.md",
  "status": null,
  "tag": "4ZTTPDXCQT",
  "uri": "m2://4ZTTPDXCQT",
  "url": "https://math2.tamnd.com/wiki/affine-differential-geometry-definition/",
  "writing": {
    "tag": "4ZTTPDXCQT",
    "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
      }
    ]
  }
}