source-code/prison_guard.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "02e353ab-f21c-4064-8a3b-ff1052f5bf3e",
   "metadata": {},
   "source": [
    "# Requirements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "79bb0672-a671-4526-ac2f-4fdd1f027e06",
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import namedtuple\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3829bfd-2e96-40c2-ac0f-7f84411559de",
   "metadata": {},
   "source": [
    "# Shapes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2328fb6-b2ea-432c-90e7-0166233b3d78",
   "metadata": {},
   "source": [
    "Simple geometric shapes are defined as `namedtuple`s, e.g., a square and a circle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "eae66591-12aa-4a42-88d6-b635706c7b14",
   "metadata": {},
   "outputs": [],
   "source": [
    "Square = namedtuple('square', ['x', 'y', 'delta_x', 'delta_y'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c18f72e8-a7d5-4665-b6ba-146309902e10",
   "metadata": {},
   "outputs": [],
   "source": [
    "Circle = namedtuple('circle', ['x', 'y', 'r'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43bfe6f9-9d5f-4384-b1d4-a0596587e193",
   "metadata": {},
   "source": [
    "# The prison"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac3541be-5831-4c77-8a17-4698c4d65ff1",
   "metadata": {},
   "source": [
    "The prison has outer walls that form a square, and a square building that is situated in the middle of the prison grounds.  The length of the walls of the outer walls is 12, that of the building's walls is 4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d59e7537-f7e7-416e-85b3-06799f430fe6",
   "metadata": {},
   "outputs": [],
   "source": [
    "walls = Square(x=0.0, y=0.0, delta_x=12.0, delta_y=12.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "eef149cb-c36c-4ccd-8d37-9ca59eb04751",
   "metadata": {},
   "outputs": [],
   "source": [
    "building = Square(x=4.0, y=4.0, delta_x=4.0, delta_y=4.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b536a9d8-869b-4aae-baed-ef1d10195235",
   "metadata": {},
   "source": [
    "# Geometries"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3f699cc-d20e-4eed-a035-3798405afd60",
   "metadata": {},
   "source": [
    "A geometry is defined as a function that takes $x$ and $y$ coordinates as parameters, and that returns `True` if that point is free, i.e., it is inside of the outer walls, but not in the building on the yard. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "ff918b9a-32ce-4fa7-b5aa-7247f482ff66",
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_square_in_square_geometry(outer_square, inner_square):\n",
    "    def is_free_coordinate(x, y):\n",
    "        is_in_outer_square = (outer_square.x <= x <= outer_square.x + outer_square.delta_x and\n",
    "                              outer_square.y <= y <= outer_square.y + outer_square.delta_y)\n",
    "        is_in_inner_square = (inner_square.x <= x <= inner_square.x + inner_square.delta_x and\n",
    "                              inner_square.y <= y <= inner_square.y + inner_square.delta_y)\n",
    "        return is_in_outer_square and not is_in_inner_square\n",
    "    return is_free_coordinate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "46d974a1-49ef-4f9f-8329-e9055856ffc8",
   "metadata": {},
   "outputs": [],
   "source": [
    "geometry = create_square_in_square_geometry(walls, building)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e552a58a-b254-4f62-8d66-3fe0e97ee5d5",
   "metadata": {},
   "source": [
    "For debugging purposes, it is convenient to have a function that displays a geometry."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "737bb2df-bf62-43ba-ac30-67e5176470d2",
   "metadata": {},
   "outputs": [],
   "source": [
    "def show_geometry(geometry, square, resolution=101):\n",
    "    x = np.linspace(square.x, square.delta_x, resolution)\n",
    "    y = np.linspace(square.y, square.delta_y, resolution)\n",
    "    X, Y = np.meshgrid(x, y)\n",
    "    shape = np.vectorize(geometry)(X, Y)\n",
    "    plt.imshow(shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "9fc0868a-425a-4e34-9a4c-59f13f7c9feb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_geometry(geometry, Square(x=-1.0, y=-1.0, delta_x=13.0, delta_y=13.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d9537040-0e3c-472c-a5f8-65406b8c2973",
   "metadata": {},
   "source": [
    "# The guard"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a832f655-283c-4409-9c6e-5bdb937054b1",
   "metadata": {},
   "source": [
    "The question is how much of the prison grounds a guard can see from a given position $x$, $y$.  First, we define a trace function that will compute the distance a guard can see when standing at $x$, $y$, looking in direction $\\alpha$, given the geometry."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "47ece0e7-d7b6-4b49-bb52-ef458371c3e8",
   "metadata": {},
   "outputs": [],
   "source": [
    "def trace(x, y, delta, alpha, geometry):\n",
    "    length = 0.0\n",
    "    delta_x, delta_y = delta*np.cos(alpha), delta*np.sin(alpha)\n",
    "    while geometry(x + delta_x, y + delta_y):\n",
    "        length += delta\n",
    "        x += delta_x\n",
    "        y += delta_y\n",
    "    return length"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d6ec4aa-43de-4d16-8526-dae6e1233a5b",
   "metadata": {},
   "source": [
    "When the guard stands in the corner, and looks towards the middle of the prison, facing the corner of the building, the distance should be $\\sqrt{32}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "153a535d-6531-4b7a-aebd-cc1f1e662014",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.6560000000002235"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trace(0.0, 0.0, 0.001, np.pi/4.0, geometry)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9c8b72b2-dca1-4eb2-8617-4471a904513d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.656854249492381"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(32.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bad42947-31a8-495e-a24d-392353a3a9de",
   "metadata": {},
   "source": [
    "When the guard stands in the corner, and looks outward, he faces the wall, so the distance should be 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "7aaa828b-e908-4729-be9f-1ac462cfa5c2",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trace(0.0, 0.0, 0.001, -np.pi/4.0, geometry)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3f9728e-e375-4c38-a1e9-7d00633b2f7e",
   "metadata": {},
   "source": [
    "The area visible to the guard can now be computed as the sum of the areas of small triangles.  The triangles are generated by letting the guard look at an angle $\\alpha$ with a field of vision of $\\Delta \\alpha$.  The area of such a triangle is approximately $R(\\alpha)^2 \\Delta\\alpha/2$, where $R(\\alpha)$ is the distance the guard can see in direction $\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "3481d5ed-2ba4-46f9-bc97-b4c6d3258e55",
   "metadata": {},
   "outputs": [],
   "source": [
    "def visible_area(x, y, geometry, delta_alpha, delta):\n",
    "    if not geometry(x, y):\n",
    "        return 0.0\n",
    "    area = 0.0\n",
    "    for alpha in np.arange(0.0, 2.0*np.pi, delta_alpha):\n",
    "        area += trace(x, y, delta, alpha, geometry)**2\n",
    "    return 0.5*area*delta_alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "4ab9b9a9-107c-465a-a1c5-26c487d5aaab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "87.98183054698586"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "visible_area(0.0, 0.0, geometry, 0.001, 0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "e03c4083-932c-47b4-afd3-92ef10a5db4d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "88.0004339404974"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "visible_area(0.0, 4.0, geometry, 0.001, 0.001)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce4b4ea1-81c6-4338-bb16-d08959bc6524",
   "metadata": {},
   "source": [
    "# Best and worst position"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81349e76-f778-4384-8902-0496918d016b",
   "metadata": {},
   "source": [
    "Where should the guard stand when his back is to one of the walls to see\n",
    "  * the largest area, and\n",
    "  * the smallest area?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "a1d5ce2c-b3f6-45df-9ebd-467a4721cdbc",
   "metadata": {},
   "outputs": [],
   "source": [
    "xs, areas = [], []\n",
    "for x in np.arange(0.0, 6.0, 0.05):\n",
    "    xs.append(x)\n",
    "    areas.append(visible_area(x, 0.0, geometry, 0.001, 0.001))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "ebfc8c60-6f20-4552-816b-2de2d1bfe5b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xs, areas);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a8227a5-32fb-4d74-ba2a-9ebd2efae469",
   "metadata": {},
   "source": [
    "The guard can survey the largest area either in the corner, or by opposite the corner of the building.  He sees the smallest area overal when facing the middle of the building, but also has suboptimal view around $x \\approx 2.5$."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}