{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and consider the [[Euclidean space|0XPR1E85W0]] $\\R^n$ with the [[Euclidean distance|0NPPGHQNXQ]] $\\norm{x - y}$.\nA *similarity transformation* of $\\R^n$ is a bijection $f \\colon \\R^n \\to \\R^n$ for which there is a real number $r \u003e 0$ such that\n$$\\norm{f(x) - f(y)} = r \\norm{x - y} \\quad \\text{for all } x, y \\in \\R^n.$$\nThe number $r$ is the *ratio* of $f$.\n\n## Remarks\n\nA similarity of ratio $1$ is an [[isometry|02HPP5VCZH]].\nEvery similarity of ratio $r$ has the form $f(x) = rAx + b$ with $A$ in the [[orthogonal group|0TKYAEJ8N7]] $\\mathrm{O}(n)$ and $b \\in \\R^n$.\nSo every similarity is a [[homothety|0ZG1PNEP35]] followed by an isometry, and the similarities of $\\R^n$ form a group.\nSimilarities preserve angles and ratios of distances.\n\nExample: $f(x) = 3x + b$ on $\\R^n$ is a similarity of ratio $3$.\nCheck: $\\norm{f(x) - f(y)} = \\norm{3x - 3y} = 3 \\norm{x - y}$.\n\nNon-example: $g(x_1, x_2) = (2x_1, x_2)$ on $\\R^2$ is a bijection.\nIt doubles the distance between $(0, 0)$ and $(1, 0)$ but keeps the distance between $(0, 0)$ and $(0, 1)$, so no single ratio $r$ works.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "D3GYHF6NAX",
    "kind": "definition",
    "msc": [
      "51M05",
      "51N20"
    ],
    "name": "Similarity transformation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Similarity_(geometry)"
      }
    ],
    "tag": "PK77W5K15V",
    "uses": [
      "0XPR1E85W0",
      "0NPPGHQNXQ",
      "02HPP5VCZH",
      "0ZG1PNEP35",
      "0TKYAEJ8N7"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/PK77W5K15V/",
  "links": [
    {
      "field": "defines",
      "target": "D3GYHF6NAX"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0NPPGHQNXQ"
    },
    {
      "field": "uses",
      "target": "02HPP5VCZH"
    },
    {
      "field": "uses",
      "target": "0ZG1PNEP35"
    },
    {
      "field": "uses",
      "target": "0TKYAEJ8N7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PK77W5K15V.md",
  "msc": [
    "51M05",
    "51N20"
  ],
  "name": "Similarity transformation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PK7/PK77W5K15V.md",
  "status": null,
  "tag": "PK77W5K15V",
  "uri": "m2://PK77W5K15V",
  "url": "https://tamnd.github.io/math2/wiki/similarity-transformation-definition/",
  "writing": {
    "tag": "PK77W5K15V",
    "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
      }
    ]
  }
}