{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $L \u003e 0$ and let $\\gamma \\colon \\R \\to \\R^2$ be an $L$-periodic curve of class $C^3$, parametrised by [[arc length|00R1JDDV5X]].\nAssume that $\\gamma$ is injective on $[0, L)$, so that it traces a simple [[closed curve|0D032NSZKY]].\nLet $k$ be the signed [[curvature|7JQS6VKFEZ]] of $\\gamma$.\nA *vertex* of $\\gamma$ is a point $s$ with $k'(s) = 0$.\nThen $\\gamma$ has at least four vertices in $[0, L)$.\n\n## Remarks\n\nThe idea is clearest for a convex curve with nonconstant $k$.\nSince $k$ is periodic, it has a maximum and a minimum, which are two vertices.\nAssume there were no others.\nThen the line through the two corresponding points of the curve meets it only there, and $k'$ has a fixed sign on each of the two arcs.\nChoose a linear function $f(x) = \\ip{x}{v} + c$ that vanishes on that line; then $k'(s) f(\\gamma(s))$ has constant sign.\nBut the plane Frenet formulas give $\\int_0^L k'(s) \\, \\gamma(s) \\, ds = 0$ and $\\int_0^L k'(s) \\, ds = 0$, so the integral of $k' \\cdot (f \\circ \\gamma)$ is $0$, a contradiction.\nThe general case of simple closed curves needs more work.\nExample: the ellipse $x^2/a^2 + y^2/b^2 = 1$ with $a \u003e b \u003e 0$ has exactly four vertices, the ends of its two axes.\nThe hypothesis that the curve is simple is needed: a limaçon with an inner loop is a closed regular curve with only two vertices.\nThe theorem was proved for convex curves by Mukhopadhyaya in 1909 and for all simple closed curves by Kneser in 1912.\n\n## Explanations\n\n### L1 human\n\nTake any smooth closed loop in the plane that does not cross itself.\nIts curvature cannot just rise once and fall once around the loop.\nThere are always at least four places where the curvature stops increasing or stops decreasing.\nAn ellipse shows that four is the best possible: it is most curved at the ends of the long axis and flattest at the ends of the short axis.\n\n### L2 human\n\nWhen $k$ never vanishes, the vertices of $\\gamma$ are exactly the points where its [[evolute|Y3ZNV24V2C]] has zero velocity.\nTypically these are cusps of the evolute, as for the four cusps of the evolute of an ellipse.\nThe proof above uses only the plane Frenet equations $T' = kN$ and $N' = -kT$, integrated over a period.\nThe theorem is an early example of a global result in differential geometry, in which a local quantity is constrained by the shape of the whole curve.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "7JQS6VKFEZ"
    ],
    "kind": "statement",
    "msc": [
      "53A04"
    ],
    "name": "Four-vertex theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q994401",
      "wikipedia": "Four-vertex theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Four-vertex_theorem"
      }
    ],
    "tag": "Z6QYDSJVFX",
    "uses": [
      "7JQS6VKFEZ",
      "0D032NSZKY",
      "06JMR2XWXW",
      "00R1JDDV5X"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/Z6QYDSJVFX/",
  "links": [
    {
      "field": "about",
      "target": "7JQS6VKFEZ"
    },
    {
      "field": "uses",
      "target": "7JQS6VKFEZ"
    },
    {
      "field": "uses",
      "target": "0D032NSZKY"
    },
    {
      "field": "uses",
      "target": "06JMR2XWXW"
    },
    {
      "field": "uses",
      "target": "00R1JDDV5X"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z6QYDSJVFX.md",
  "msc": [
    "53A04"
  ],
  "name": "Four-vertex theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z6Q/Z6QYDSJVFX.md",
  "status": null,
  "tag": "Z6QYDSJVFX",
  "uri": "m2://Z6QYDSJVFX",
  "url": "https://math2.tamnd.com/wiki/four-vertex-theorem/",
  "writing": {
    "tag": "Z6QYDSJVFX",
    "kind": "statement",
    "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
      }
    ]
  }
}