{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "HDD0EB48AB"
    }
  ],
  "body": "\n## Summary\n\nThe Hilbert metric is a distance on any bounded open [[convex set|02AS45T3H9]] of $\\R^n$.\nIt is the logarithm of a [[cross-ratio|09ASMRMD67]] of the two points and the two boundary points on the line through them.\nIt turns the convex set into a [[metric space|0VKD1TP7F3]] in which straight segments are shortest paths.\nWhen the convex set is a ball, it is the [[Beltrami–Klein model|1WN9B18H86]] of hyperbolic space, so Hilbert geometries generalize the hyperbolic [[Cayley–Klein metric|J7ZE3EGRYX]].\n\n## Explanations\n\n### L1 human\n\nTake a convex region, such as a disc or a triangle.\nTo measure the distance between two points, extend the segment between them until it hits the boundary on both sides.\nThe distance compares how close each point is to each boundary point, through a logarithm.\nNear the boundary, distances blow up, so the boundary is infinitely far away.\nIn a disc this gives a picture of the hyperbolic plane in which lines look straight.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "51F99",
      "52A20",
      "53C60"
    ],
    "name": "Hilbert metric",
    "same_as": {
      "wikidata": "Q5761213",
      "wikipedia": "Hilbert metric"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert_metric"
      }
    ],
    "specializes": [
      "0VKD1TP7F3"
    ],
    "tag": "P923DHV74J"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/P923DHV74J/",
  "links": [
    {
      "field": "specializes",
      "target": "0VKD1TP7F3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/P923DHV74J.md",
  "msc": [
    "51F99",
    "52A20",
    "53C60"
  ],
  "name": "Hilbert metric",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P92/P923DHV74J.md",
  "status": null,
  "tag": "P923DHV74J",
  "uri": "m2://P923DHV74J",
  "url": "https://tamnd.github.io/math2/wiki/hilbert-metric/",
  "writing": {
    "tag": "P923DHV74J",
    "kind": "concept",
    "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
      }
    ]
  }
}