{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\gamma \\colon I \\to \\R^2$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ on an interval $I$, and let $t_0 \\in I$.\nLet $T$ be its unit [[tangent vector|0D98771SM0]], let $N$ be $T$ rotated by a right angle counterclockwise, and let $k$ be its signed [[curvature|7JQS6VKFEZ]].\nAssume that $k(t_0) \\ne 0$.\nThe *centre of curvature* at $t_0$ is $c = \\gamma(t_0) + N(t_0) / k(t_0)$, and the *radius of curvature* is $1 / \\abs{k(t_0)}$.\nThe *osculating circle* of $\\gamma$ at $t_0$ is the [[circle|045DN06F3S]] with centre $c$ and radius $1 / \\abs{k(t_0)}$.\n\n## Remarks\n\nExample: for $\\gamma(t) = (t, t^2)$ at $t_0 = 0$, one has $T = (1, 0)$, $N = (0, 1)$ and $k(0) = 2$.\nSo the osculating circle has centre $(0, 1/2)$ and radius $1/2$.\nNon-example: for $\\gamma(t) = (t, t^3)$ at $t_0 = 0$, the curvature is $0$, so there is no osculating circle there; the tangent line plays its role.\nThe osculating circle passes through $\\gamma(t_0)$, has the same tangent there, and is the only circle with contact of order at least $2$ with the curve.\nFor a curve in $\\R^3$ with $\\kappa(t_0) \u003e 0$, the osculating circle is defined in the same way, with $N$ the principal normal.\nIt lies in the osculating plane, spanned by $T$ and $N$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "NWSFMCRWHJ",
    "kind": "definition",
    "msc": [
      "53A04"
    ],
    "name": "Osculating circle",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Osculating_circle"
      }
    ],
    "tag": "Z2Z9BE946H",
    "uses": [
      "7JQS6VKFEZ",
      "06JMR2XWXW",
      "0D98771SM0",
      "045DN06F3S"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Z2Z9BE946H/",
  "links": [
    {
      "field": "defines",
      "target": "NWSFMCRWHJ"
    },
    {
      "field": "uses",
      "target": "7JQS6VKFEZ"
    },
    {
      "field": "uses",
      "target": "06JMR2XWXW"
    },
    {
      "field": "uses",
      "target": "0D98771SM0"
    },
    {
      "field": "uses",
      "target": "045DN06F3S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z2Z9BE946H.md",
  "msc": [
    "53A04"
  ],
  "name": "Osculating circle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z2Z/Z2Z9BE946H.md",
  "status": null,
  "tag": "Z2Z9BE946H",
  "uri": "m2://Z2Z9BE946H",
  "url": "https://math2.tamnd.com/wiki/osculating-circle-definition/",
  "writing": {
    "tag": "Z2Z9BE946H",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}