{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "428R02EYWS"
    },
    {
      "field": "uses",
      "target": "FX40F4A63V"
    },
    {
      "field": "defines",
      "target": "VNXJRNE33W"
    }
  ],
  "body": "\n## Summary\n\nTwo points on a [[geodesic|NSYMQMHPDN]] are conjugate if a nonzero [[Jacobi field|9AHMM41PJ3]] along the geodesic vanishes at both of them.\nRoughly, a family of geodesics leaving the first point refocuses at the second.\nThey are exactly the critical values of the [[exponential map|Y7Z0AGY1A7]] along the geodesic.\nA geodesic stops being length-minimizing after it passes its first conjugate point.\n\n## Explanations\n\n### L1 human\n\nSend out geodesics from a point in nearly the same direction.\nIf, further on, they come back together to first order at some point, that point is conjugate to the starting point.\nOn the unit sphere, all geodesics from the north pole meet at the south pole, so the poles are conjugate along every meridian.\nIn the flat plane, straight lines from a point never meet again, so there are no conjugate points.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "53C22"
    ],
    "name": "Conjugate points",
    "same_as": {
      "wikidata": "Q5161158",
      "wikipedia": "Conjugate points"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Conjugate_points"
      }
    ],
    "tag": "ZPXE5VSZ35"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZPXE5VSZ35/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ZPXE5VSZ35.md",
  "msc": [
    "53C22"
  ],
  "name": "Conjugate points",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZPX/ZPXE5VSZ35.md",
  "status": null,
  "tag": "ZPXE5VSZ35",
  "uri": "m2://ZPXE5VSZ35",
  "url": "https://math2.tamnd.com/wiki/conjugate-points/",
  "writing": {
    "tag": "ZPXE5VSZ35",
    "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
      }
    ]
  }
}