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  "body": "\n## Problem\n\nTwo players move in the plane.\nThe pursuer $P$ is a car: it moves with constant speed $w_1 \u003e 0$ and has minimum turning radius $R \u003e 0$.\nIts position $(x_1, y_1)$ and heading $\\theta$ obey\n$$x_1' = w_1 \\sin \\theta, \\qquad y_1' = w_1 \\cos \\theta, \\qquad \\theta' = \\frac{w_1}{R} \\phi, \\qquad \\phi(t) \\in [-1, 1].$$\nThe evader $E$ moves with speed at most $w_2$, where $0 \u003c w_2 \u003c w_1$, in any direction: its position $(x_2, y_2)$ obeys $x_2' = w_2 \\sin \\psi$, $y_2' = w_2 \\cos \\psi$ with $\\psi(t)$ free.\nLet $\\ell \u003e 0$ be the capture radius; capture happens at the first time the distance between $P$ and $E$ is at most $\\ell$.\nThe problem is a [[differential game|THSHGNJNPS]] between $P$ and $E$.\nIt asks for which parameters $w_1, w_2, R, \\ell$ and initial positions the pursuer can force capture in finite time against every play of the evader.\nIt also asks for the optimal strategies when the pursuer minimises and the evader maximises the time to capture.\n\n## Remarks\n\nIsaacs used the problem as a model example in his theory of differential games.\nBy using coordinates attached to the car, the game reduces to a two-dimensional state, the position of the evader relative to the pursuer.\nThe evader is slower but more agile, so the car cannot simply drive at it: when it comes close, the evader can step aside inside the turning circle.\nIsaacs found an explicit condition on the ratios $w_2 / w_1$ and $\\ell / R$ under which capture is possible from every initial position.\nThe optimal strategies of the time-to-capture game turned out to involve several kinds of singular curves in the reduced state plane.\n\n## Explanations\n\n### L1 human\n\nA driver in a car tries to run over a pedestrian in a large empty parking lot.\nThe car is faster, but it cannot turn sharply, while the pedestrian can change direction at will.\nThe question is when the driver can be sure to succeed, and how both should move.\nIt is a classical test problem for the theory of pursuit and evasion.\n",
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    "history": {
      "first_stated": "1951",
      "posed_by": "Rufus Isaacs"
    },
    "kind": "problem",
    "msc": [
      "49N75",
      "91A24",
      "91A23"
    ],
    "name": "Homicidal chauffeur problem",
    "same_as": {
      "wikidata": "Q5890773",
      "wikipedia": "Homicidal chauffeur problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homicidal_chauffeur_problem"
      }
    ],
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    "uses": [
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  "link": "https://tamnd.github.io/math2/node/M0B618E2P6/",
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  "markdown": "https://tamnd.github.io/math2/node/M0B618E2P6.md",
  "msc": [
    "49N75",
    "91A24",
    "91A23"
  ],
  "name": "Homicidal chauffeur problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M0B/M0B618E2P6.md",
  "status": null,
  "tag": "M0B618E2P6",
  "uri": "m2://M0B618E2P6",
  "url": "https://tamnd.github.io/math2/wiki/homicidal-chauffeur-problem/",
  "writing": {
    "tag": "M0B618E2P6",
    "kind": "problem",
    "level": "W0",
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        "note": "the row of the inventory that the node accounts for",
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