{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $Q$ be a nondegenerate [[quadric|Q3Q4FF23VG]] in the real [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^n$, called the *absolute*.\nLet $c$ be a nonzero real or imaginary constant.\nLet $x$ and $y$ be two distinct points not on $Q$ such that the line $xy$ is not tangent to $Q$.\nLet $a$ and $b$ be the two points where the line $xy$ meets $Q$, taken in the complexification when they are not real.\nThe *Cayley–Klein distance* of $x$ and $y$ is\n$$d(x, y) = c \\log (x, y; a, b),$$\nwhere $(x, y; a, b)$ is the [[cross-ratio|09ASMRMD67]] of the four points on the line.\nAlso $d(x, x) = 0$.\n\n## Remarks\n\nHyperbolic case: let $Q$ be an ellipsoid and let $x$ and $y$ lie inside it.\nThen $a$ and $b$ are real.\nIn an affine chart with the points in the order $a, x, y, b$ on the line, take $c = \\frac{1}{2}$ and the cross-ratio $\\frac{\\abs{xb} \\, \\abs{ya}}{\\abs{xa} \\, \\abs{yb}}$.\nThis ratio is greater than $1$, so $d(x, y) \u003e 0$, and $d(x, y) \\to \\infty$ as $y$ tends to $b$.\nThis is the [[Beltrami–Klein model|1WN9B18H86]] of hyperbolic space with curvature $-1$.\n\nElliptic case: if $Q$ has no real points, such as $x_0^2 + x_1^2 + x_2^2 = 0$, then $a$ and $b$ are complex conjugate, and an imaginary constant $c$ gives the real distance of [[elliptic geometry|H5GB19W2H3]].\n\nNon-example: a degenerate quadric, such as a double plane, does not fit the definition, because a line meets it in one point only.\nEuclidean geometry appears only as a limit of the nondegenerate cases.\n\nConventions for the order in the cross-ratio differ between sources; they change the sign of the logarithm, not the geometry.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "J7ZE3EGRYX",
    "kind": "definition",
    "msc": [
      "51F99",
      "51M10"
    ],
    "name": "Cayley–Klein metric",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cayley%E2%80%93Klein_metric"
      }
    ],
    "tag": "V50N3JEBDP",
    "uses": [
      "ZT9A6JK8QF",
      "Q3Q4FF23VG",
      "09ASMRMD67"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/V50N3JEBDP/",
  "links": [
    {
      "field": "defines",
      "target": "J7ZE3EGRYX"
    },
    {
      "field": "uses",
      "target": "ZT9A6JK8QF"
    },
    {
      "field": "uses",
      "target": "Q3Q4FF23VG"
    },
    {
      "field": "uses",
      "target": "09ASMRMD67"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/V50N3JEBDP.md",
  "msc": [
    "51F99",
    "51M10"
  ],
  "name": "Cayley–Klein metric",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V50/V50N3JEBDP.md",
  "status": null,
  "tag": "V50N3JEBDP",
  "uri": "m2://V50N3JEBDP",
  "url": "https://tamnd.github.io/math2/wiki/cayley-klein-metric-definition/",
  "writing": {
    "tag": "V50N3JEBDP",
    "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
      }
    ]
  }
}