{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit [[normal vector|0BJDQ2HBH1]] field $N$.\nLet $\\alpha \\colon I \\to S$ be a curve of class $C^2$ parametrised by [[arc length|00R1JDDV5X]] on an interval $I$.\nThe *geodesic curvature* of $\\alpha$ at $s \\in I$ is\n$$\\kappa_g(s) = \\ip{\\alpha''(s)}{N(\\alpha(s)) \\times \\alpha'(s)},$$\nwhere $\\times$ is the [[cross product|0SEDGXG9S3]].\nThe *normal curvature* of $\\alpha$ at $s$ is $\\kappa_n(s) = \\ip{\\alpha''(s)}{N(\\alpha(s))}$.\n\n## Remarks\n\nSince $\\alpha'$, $N \\times \\alpha'$ and $N$ form an orthonormal basis and $\\ip{\\alpha''}{\\alpha'} = 0$, one has $\\alpha'' = \\kappa_g \\, N \\times \\alpha' + \\kappa_n N$.\nSo $\\kappa^2 = \\kappa_g^2 + \\kappa_n^2$, where $\\kappa$ is the [[curvature|7JQS6VKFEZ]] of $\\alpha$ as a space curve.\nExample: on the unit sphere with the outer normal $N(p) = p$, a great circle $\\alpha$ satisfies $\\alpha'' = -\\alpha$, which is normal to the sphere, so $\\kappa_g = 0$.\nA circle of latitude at polar angle $\\varphi_0 \\in (0, \\pi)$ has $\\kappa = 1 / \\sin \\varphi_0$ and $\\abs{\\kappa_g} = \\abs{\\cot \\varphi_0}$.\nNon-example: for a curve that is not of unit speed, the formula above is not the geodesic curvature; one must first reparametrise by arc length.\nFor a curve in the $xy$-plane with $N = (0, 0, 1)$, the geodesic curvature is the signed curvature of the plane curve.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "550P8PFMVC",
    "kind": "definition",
    "msc": [
      "53A05"
    ],
    "name": "Geodesic curvature",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Geodesic_curvature"
      }
    ],
    "tag": "1399T2Q2ZS",
    "uses": [
      "6080X3FAD4",
      "7JQS6VKFEZ",
      "0BJDQ2HBH1",
      "0SEDGXG9S3",
      "00R1JDDV5X"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/1399T2Q2ZS/",
  "links": [
    {
      "field": "defines",
      "target": "550P8PFMVC"
    },
    {
      "field": "uses",
      "target": "6080X3FAD4"
    },
    {
      "field": "uses",
      "target": "7JQS6VKFEZ"
    },
    {
      "field": "uses",
      "target": "0BJDQ2HBH1"
    },
    {
      "field": "uses",
      "target": "0SEDGXG9S3"
    },
    {
      "field": "uses",
      "target": "00R1JDDV5X"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1399T2Q2ZS.md",
  "msc": [
    "53A05"
  ],
  "name": "Geodesic curvature",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/139/1399T2Q2ZS.md",
  "status": null,
  "tag": "1399T2Q2ZS",
  "uri": "m2://1399T2Q2ZS",
  "url": "https://math2.tamnd.com/wiki/geodesic-curvature-definition/",
  "writing": {
    "tag": "1399T2Q2ZS",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}