Tutorial.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "\n",
    "\n",
    "The `tyssue` library seeks to provide a unified interface to implement\n",
    "bio-mechanical models of living tissues. It's main focus is on **vertex based epithelium models**.\n",
    "\n",
    "\n",
    "### Outline\n",
    "\n",
    "1. Core objects and datastructures\n",
    "\n",
    "2. Visualisation\n",
    "\n",
    "3. Quasistatic models\n",
    "\n",
    "4. Event management\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[tyssue docs](https://tyssue.readthedocs.io/en/latest/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import ipyvolume as ipv\n",
    "\n",
    "\n",
    "# Core objects and geometry\n",
    "\n",
    "from tyssue import (\n",
    "    Monolayer,\n",
    "    ClosedMonolayerGeometry,\n",
    "    Sheet,\n",
    "    ClosedSheetGeometry\n",
    ")\n",
    "\n",
    "# Generation tools\n",
    "from tyssue import generation\n",
    "\n",
    "# Viz\n",
    "from tyssue.draw import sheet_view\n",
    "\n",
    "# Dynamical model\n",
    "from tyssue.dynamics import model_factory, effectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data structure\n",
    "\n",
    "\n",
    "![Tyssue data structure](data/svg/data_model.svg)\n",
    "\n",
    "\n",
    "Two main class of models: 2D in 2D or 3D (`Sheet`) and full 3D, multilayer (`Epithelium`) or monolayer (`Monolayer`).\n",
    "\n",
    "There are three (resp. 4) datasets, representing vertices, edges, faces (and cells).\n",
    "\n",
    "The main idea is to do the computation on the `edge` dataset, as it retains a reference to the indices of its source (`srce`) and target (`trgt`) vertices, its `face` (and its `cell`). There are mechanisms to copy the data belonging to a vertex or a face to the corresponding edges, called upcasting. \n",
    "\n",
    "The edges are actually half-edges in 2D and in 3D the data structure is that of a Linear Cell Complex (CGAL).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "109 ms ± 7.43 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit ClosedMonolayerGeometry.update_all(mono)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "182\n"
     ]
    }
   ],
   "source": [
    "mono = generation.spherical_monolayer(10, 12, 100)\n",
    "ClosedMonolayerGeometry.update_all(mono)\n",
    "print(mono.Nc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "apical_sheet = mono.get_sub_sheet(\"apical\")\n",
    "apical_sheet.reset_index()\n",
    "apical_sheet.reset_topo()\n",
    "\n",
    "ClosedSheetGeometry.update_all(apical_sheet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/guillaume/miniconda3/lib/python3.7/site-packages/numpy/core/fromnumeric.py:2223: FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.\n",
      "  return ptp(axis=axis, out=out, **kwargs)\n"
     ]
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "352ac7f5f364461885981da8ab240ee7",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Figure(camera=PerspectiveCamera(fov=46.0, position=(0.0, 0.0, 2.0), quaternion=(0.0, 0.0, 0.0, 1.0), scale=(1.…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ipv.clear()\n",
    "fig, mesh = sheet_view(mono, mode=\"3D\", face={\"visible\": False})\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each dataset can be accessed as a value of the `mono.datasets` dictionnary or through `mono.edge_df` (or `mono.face_df`, etc)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict_keys(['cell', 'vert', 'edge', 'face'])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mono.datasets.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "from tyssue.io import obj\n",
    "\n",
    "obj.save_junction_mesh(\"test.obj\", mono)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
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       "      <th>3</th>\n",
       "      <td>3.096127</td>\n",
       "      <td>6.712052</td>\n",
       "      <td>10.540596</td>\n",
       "      <td>1</td>\n",
       "      <td>57.239452</td>\n",
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       "      <td>7</td>\n",
       "      <td>30</td>\n",
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       "      <th>4</th>\n",
       "      <td>4.736138</td>\n",
       "      <td>10.555625</td>\n",
       "      <td>7.683949</td>\n",
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       "      <td>90.907471</td>\n",
       "      <td>-51.235093</td>\n",
       "      <td>9</td>\n",
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       "             x          y          z  is_alive       area        vol  \\\n",
       "cell                                                                   \n",
       "0    -1.771128  -5.117349  -8.773759         1  63.032706 -32.217097   \n",
       "1    -0.498756  -7.532178  -5.408639         1  53.894137 -26.609051   \n",
       "2    -4.090607  -6.606667  -6.002684         1  46.031622 -20.828427   \n",
       "3     3.096127   6.712052  10.540596         1  57.239452 -27.767288   \n",
       "4     4.736138  10.555625   7.683949         1  90.907471 -51.235093   \n",
       "\n",
       "      num_faces  num_ridges  \n",
       "cell                         \n",
       "0             8          36  \n",
       "1            10          48  \n",
       "2             7          30  \n",
       "3             7          30  \n",
       "4             9          42  "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mono.datasets['cell'].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "      srce  trgt  face        dx        dy        dz        nx        ny  \\\n",
       "edge                                                                       \n",
       "0       28     0     0 -1.864921 -0.287282  0.634712 -0.497366 -2.982764   \n",
       "1      170   169   129 -0.795092 -0.124975  0.446275  0.257935 -0.625810   \n",
       "2       78    79    65  1.135073  1.388797 -0.709251  2.860254 -0.706708   \n",
       "3      295   170   129 -0.021509  0.396630  0.897120  0.254224 -0.614782   \n",
       "4       84    78    65  1.567257  0.393904 -1.146787  2.717778 -0.866624   \n",
       "\n",
       "            nz        sx  ...  uy  uz  edge_o  srce_o  trgt_o  face_o  cell_o  \\\n",
       "edge                      ...                                                   \n",
       "0    -2.811420 -0.166006  ...   0   0       0      28       0       0       0   \n",
       "1     0.284291  4.162625  ...   0   0    1560     170     169     129       0   \n",
       "2     3.193686  7.250890  ...   0   0     790      78      79      65       0   \n",
       "3     0.277900  4.184134  ...   0   0    1562     295     170     129       0   \n",
       "4     3.416579  5.683633  ...   0   0     788      84      78      65       0   \n",
       "\n",
       "      sub_area  is_valid  segment  \n",
       "edge                               \n",
       "0     2.064482      True    basal  \n",
       "1     0.367080      True    basal  \n",
       "2     2.172563      True    basal  \n",
       "3     0.360491      True    basal  \n",
       "4     2.225442      True    basal  \n",
       "\n",
       "[5 rows x 36 columns]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mono.edge_df.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 314,
   "metadata": {},
   "outputs": [],
   "source": [
    "from tyssue import Sheet, SheetGeometry\n",
    "from tyssue.generation import generate_ring\n",
    "\n",
    "\n",
    "sheet = generate_ring(24, 12, 18)\n",
    "\n",
    "PlanarGeometry.update_all(sheet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 324,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Shape of rotation matrices (2, 2, 96)\n"
     ]
    },
    {
     "ename": "KeyError",
     "evalue": "'dz'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m~/miniconda3/lib/python3.7/site-packages/pandas/core/indexes/base.py\u001b[0m in \u001b[0;36mget_loc\u001b[0;34m(self, key, method, tolerance)\u001b[0m\n\u001b[1;32m   2656\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2657\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2658\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'dz'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-324-315fa065dc0f>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m     20\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     21\u001b[0m \u001b[0;31m# φ is the angle we want\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 22\u001b[0;31m \u001b[0msheet\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0medge_df\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m\"phi\"\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marctan2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msheet\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0medge_df\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'dx_r'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msheet\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0medge_df\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'dz'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     23\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/lib/python3.7/site-packages/pandas/core/frame.py\u001b[0m in \u001b[0;36m__getitem__\u001b[0;34m(self, key)\u001b[0m\n\u001b[1;32m   2925\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnlevels\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2926\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_getitem_multilevel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2927\u001b[0;31m             \u001b[0mindexer\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcolumns\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2928\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mis_integer\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mindexer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2929\u001b[0m                 \u001b[0mindexer\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mindexer\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/lib/python3.7/site-packages/pandas/core/indexes/base.py\u001b[0m in \u001b[0;36mget_loc\u001b[0;34m(self, key, method, tolerance)\u001b[0m\n\u001b[1;32m   2657\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2658\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2659\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_engine\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_loc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_maybe_cast_indexer\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2660\u001b[0m         \u001b[0mindexer\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_indexer\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmethod\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtolerance\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtolerance\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2661\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mindexer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mindexer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msize\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/index.pyx\u001b[0m in \u001b[0;36mpandas._libs.index.IndexEngine.get_loc\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;32mpandas/_libs/hashtable_class_helper.pxi\u001b[0m in \u001b[0;36mpandas._libs.hashtable.PyObjectHashTable.get_item\u001b[0;34m()\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'dz'"
     ]
    }
   ],
   "source": [
    "\n",
    "## Angle θ of the source in the cylindrical coordinate system\n",
    "theta = np.arctan2(sheet.edge_df[\"sy\"], sheet.edge_df[\"sx\"],)\n",
    "cost, sint = np.cos(theta), np.sin(theta)\n",
    "\n",
    "# One rotation matrix per edge\n",
    "rot_mat = np.array(\n",
    "    [[cost, sint],\n",
    "     [-sint, cost]]\n",
    ")\n",
    "print('Shape of rotation matrices', rot_mat.shape)\n",
    "\n",
    "# We want to get the edge coordinates w/r to θ\n",
    "dx_dy = sheet.edge_df[['dx', 'dy']].to_numpy()\n",
    "sheet.edge_df['dx_r'] = sheet.edge_df['dx'].copy()\n",
    "sheet.edge_df['dy_r'] = sheet.edge_df['dy'].copy()\n",
    "\n",
    "# numpy einsum magic (need lots of trial and error :/)\n",
    "sheet.edge_df[['dx_r', 'dy_r']] = np.einsum('jik, ki-> kj', rot_mat, dx_dy)\n",
    "\n",
    "# φ is the angle we want\n",
    "sheet.edge_df[\"phi\"] = np.arctan2(sheet.edge_df['dz'], sheet.edge_df['dx_r'])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 317,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 2, 96)"
      ]
     },
     "execution_count": 317,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rot_mat.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.scatter(sheet.edge_df['dx_r'], sheet.edge_df['dy_r'], alpha=0.8)\n",
    "\n",
    "ax.set_aspect(\"equal\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "metadata": {},
   "outputs": [],
   "source": [
    "phi = np.arctan2(sheet.edge_df['dy_r'], sheet.edge_df['dx_r'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "metadata": {},
   "outputs": [],
   "source": [
    "sheet.face_df['theta'] = np.arctan2(sheet.face_df['y'], sheet.face_df['x'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Nqa8bEI6dYzsmE6uH/q6mYY3n8WcDoCVOZ3Y2bh3yWKEeOx0+uQbJnTxnWLbFI/0D30sEwdDwVyLvpbezCTevIlADkZCs8m8ijuHwnMgC4YjDCRiKMAAYbg969m0AuIRpp/j2/Jp4fPh34sOZuuhjwKFCMF21qXFMJgl2DT3JqjNREyLpjBxumnfj1l8ChyIfUxf/w7DGey9SWoSbfwhKoaRAIVCea/GewKcxUQXCEYfjMFRhSJJfNptg5Aw0AS7s0m+de49eng4HTWiEQlkIDXQkHk8IlQoHaOJGqan9Qx4i1r8XSN1R7UDgwMA5ChAYLm9Kxk3SvPWbgP3WLaURDJVTNOPkti0TUSAccTgGwxWGJJVnfBLTNNA1haHsD23Nuv9Jqa1FM20fRzKsuX/z71M2ts/XPeTXugy7TV8qjmg07Xo1MZYtNK7c1K4aajc+juGKoR3mz6g88wunNMZEEwhHHAYhVcKQpGSB7X8wEv6HQKCWg/veOcGrTh5/ll0TIumY9Hs2n+AVxycYLEDpgKaGFbNwuSO0x7w0RrOIx93DssklX0yco7D/ZPMmn3zdhhPR2bwHv2cdGoeEIa/qO0MaayIJhFN9+j2kWhgAhKbjK/pXwi0/wi1MIsqFiP6RSGg2Xn9OSuw2tY9gyOdwIYmh09m445Ra2nWGQry2v5adbW3M9uZwlreTd9pK2NhRgqv9tSHZtCCjhHUdxZiWjrI8uHtfO9S3M+PY7fDeS7B74PQ/CkE4NvxzJEnMeIR45y9BKaSyz6PqWbdiuIdedyIpED9fv46wiOJTngGBGE9nMYRSY6eEwrJly9T69etHbf50CMPh1G97Aa+2AksJYtjOuaK5qWt337Iz2eLeHrtw3rHHjpomqxvqeauh/ojHldLwyii3VbzAWwfLeLFhJlmZQ4sKZMutHIx6QIDuLkLzuAZqVCY5u7SU86eU43cfe2VRv+FbeL39WJbAQqNgzn1oKapD2bLjTgCEtH0ZoehCKpbekpKxO0Mhfr5+HQA+5cFAZ0Fh0agLhBBig1LqhArrrBwSpFsYAKYuuIL96zbg83djSBNTGNSuv5fKZalxUvb3TyUjox4diYWGGQtjuO2O00optrUcZEVtLcF47LBXCaTlwrLcWKYbKV2EEfyp+n3EzF48xmRQQ/sziVKJkG2EzWxyPZNQMQVCorQ4SjNRmsWapibWNNnhV5emcWnlNJaUTB7oJaGkxOvtT5yj0IlGM1ImDDXrfknAbyc6xZSOQqRMGGD8ryCclQMjIwxJlFS07boLBcSkjhQ6cS6jbO4Hhz12PBams/qbSAkx5aKmZwa7/WfT0Nt7xHVS6limB2l5sCwX73U93bxwEVfNmoXXSE3Fo5hl8dfqvfx2yyZi1qHzmQoFwkyIRbIp7iGKAgHOie+kKHMrSkJc6WSUfplAzvA7ZrXUrIfwY2hKEU+cm5g0+/toeurvl2NtBXGyK4fTXhx+svpNXt2/H5ceTbswJAn1dtDf8D17C6AMlNAIlNxJRu7wezge3HknjWEff2iZB0Lg8k1BKR3LdGNZbqTlRqkjO06/r7KSTy5YSGHg5P0Aw6E7EuaJ7dsHKnInURxaVaBZKCG52Psa7wYnYSmDy/LqmLLgvmHPHw310V17DwKQll2Y1lv4BbIKyoc99rF4r0DoSrAgL5fvfejDaZvzWDjicBI8ufo5fr5zHV4jRkEghC9QkHZhSNK89y302FNIJYgqA4SgYPYP0LSht4r/W+0+3t27DuXqoq4vj6jpwvDORckjVwCzJ03i1sVLmFtQONy3kRL2dXby2y2b2HTwyMNjSpho4W30CRdS04BszqiYyUdmzhpWda1DfgYwlUbEuoipC68ezls4KZICsSC4kRbNSzDu4SNl8/jHKz6V9rkPx/E5nICYZfFc3Wq8hqIwqwtdKD6SuYJi39kjMn9J1fnUrFlPRkY9rkRFpwPb7qZs0fdOeayark4e3W5nI0pPKRnNMfqVn1jcRZbbIMfr5ZZFi7mkctqY7MMwPS+Pf7/kUsD2jaxuaOCRLZs42N9PbsQkrHsIWx58+TlsaTnIlpaDXDt7LvMKT13catZ+n0AAezuhdCLRTKYuTb8wAOR4PHwm5y80+L1saJ+JB8mjtQ3cEImQ7U1tQlcqOG1XDp9/4Tmaew6wMOd1/K4wC70dFLrDAOjZnyG/bPiHhE6G1sRdLCZ1LKETjp1FxeLrT+q1vdEo/712NWBHGcLRTCKxTJRSZJlR/uuqj1OSnZpQ6WjREwryLy/+hY5oHCXAEjHiWhyViHrcsews8k+yW3bTrhUY8gXbAZnIZyiY81/HPDeRSnrb9hNp/QkAUQsebFyKrhR+owyPN8CzN96UdhuSnOzKIZ29MvOEEK8IIfYm/h0zTW2a+/qo7+3BcEFcK+T9Jfsp9/bYNRgAq+fX1G343YjYkjvDTrZJHtDyudfSdXDvcV9jKclvt2waEIZoPEB3fzGRWBYg+OFlV/Lop24b98IAkO0P8L8fu55ffPgqBAJDefBaAQzpQiD4+fq1/HzdWuLW8VPGg90tGPKFIw5UZU752ogIQ/3W52xhUAohFW40SjxBlFGI5rFXDDvbWtNux6mSzl6ZXwVWKKWqgBWJn8cE//j8M4CiwN9KiAwKS79LPO7C0Cw8xO0sQ+9mDm77cmrOKRwHl8ePlnmrXSBGmKAUsc4HsMzB+zW+09jA91a9zv7ebkzLQ09/EcFwHkrp3L50Gc/ecDNV+flptXk0KM3K4tkbb+JL552PQOCWXjymD13pdIRD/Odbb7LiGN2ppFT0N/3ArkiVKNxiGVfhy0pN8ZljoaRi//p78OqvIVAIBTLhDP7MebdguL1EVBylFF999ZW02jIUUiIOSqk3gM73PHw18Eji+0eA9FUAPQVW1dsHibI93bj1KAsKi/C5PZQu+r59SEpTeISJjkTTLNp2fZm+9vSUeksyaeoCguEFiQNathi17/n6Edc09vXynTdX8kptNVLq9Ifz6A0WYkk3S4tLePr6m/jwzNSlFI9VLiyv4JkbPsmlldPQ0PFYfjyWF01prG5s4DtvvEZ155F/io2b7fJ9GvaBqlBoMiWzLk6rneG+Dtp23YXP14emJNLSkEqnPzSdwnn343b7uKRyGgpFNHEo7487xlZrl5T5HBKNdP+ilJqf+LlbKZVz2PNdSqnjbi3S7XNQSnH1E4+iCYvyrBp0zeIbFyw/wvPd0bQLq+dXAMSljomt9FF1CVPmpzfs1LL9LoRQ9rxCJxiaRfGiv+PH76zGQoIShGOZhKOZJHX9tx/9GDleX1rtGquE4nFue/Zp+mJ2l2xTxDD1OMkTIf989rl07noen+stNBRxy/5dHi9zNBUc2P0GhvkMKNAPO6+hfNdTNO2sI6797huvAZAhvGhC8NT1Nx6z03qqGFGfw3AYyV6ZP1u3BoA8bzu6ZvHRWXOOConll845olCLW5gIwCNWUr/hntR2gXoPBbPtcvaGZiGkZFU4yvdffxkLSSzup7u/mHA0G9D47iXv59kbbj5thQHA73Lx6Mc+zo8/dDkCgUt58Jp+DGkH4R5e+xJWxhrbz2DZf+q50/89rTbVrPkhhvkMAoWmGBCGrIq7jxIGgFsWLQEgouyu618bQ9uLdIpDS7IlXuLfQT0uSqkHlVLLlFLLCgpOvuP0qRKOx3m5phq3FiXbYx9Dnl9YPOi1hstL4dwf0t8/A11IPCKOphRebx9tu+4i1NuRFhs1w8Az6f+BgLWRIvZGsombPfQGC+kP5yOVwY3zF/LsDTezsGhw209HpuXm8eyNN/HZZWci0HBLH17TxyRjP7+pm8+vDiwgLF2IwM24vOlJ9IpHgrTuuJOMjCY07GIxdk2ISRTOux9vYHDn8JRsu/KWiYWpJLs72mkLpq7D+XBIpzg8CyQT1W8BnknjXCfk9r88DcAkfytCKD53xonzGaaddQfScwNCKDx6HAMLBfQ3fI/mPW+mxU7/pCms7JvNgWgG4YiXzr5M4qaL6bl5PPnxG7lx/sITD3KackXVTJ6+4ZMsmzwZDYO23gz0mCBkuXmo6UwySuenZd62/Vvo2mf7NXQlsSwNpTRi2oeoPPPExYW/eO55AAPOyc88+3Ra7DxVUhXKfAx4G5glhGgUQnwG+E/gA0KIvcAHEj+PCjVdnfREo/hd/fhdQXI8vpOOjRdPP4usikP9KLyYCAV6/Glq1/0opXbGLYvvv/Um1dEq+kMl+KSGV2Tw0Eeu5UcfvAKXPvTsydMFTQj+7eJL+N01HyMgclFxAxHLwu2fxP1vv0U4Hk/pfLXrHkb1/9beRkiFKQ1AECj9EmVzPnBSY/hdbqry85FI4gmH9PoDw6sFmgpOiySoqx7/P0BSnlWHS4/xlfMuGtIHrXbtfxDIaEMqQVwZA+XPcqd/G5d3eB20k8IA0BPOpz9mLzefuf6mlDbiPd24/k+PEzZN4phEsE+j3nnu+fhcwztUJs047bvt6LyGwrIEoBGLeSld9G3EKabBK6W4983XEQgyhAchRNoSo8aNQzLd/LXaTijK8XTj0mOcXTplyHfgyrO+RowPoQmFW5gYyaKu+/6N9votQ7bREYb08cR1N+AzDFwYeLFrRgx3BdHdUjMgDMltBGiE4+dQtuTeUxYGACEEV1TNPCK0+euNG4ZsYyqY0OIgleLn69egC5M8XzsAH5g2Y1hjls29jEDJlxBC4dJM3IlfpOz7LbXrHz7l8RxhSD+pFIi6zU8Sa/8ZoBL1Ju1thGfSHZQv/viw7FxaYp/KjSkTqSTP7H6XqGkOa8zhMKHF4T9WvQ5Avq8dTUhumJcaZ14gt4SC2fcRjfrQNWlnVSpFwLeD1h13Is2T+6NzhGHkGK5AKKlo2PhN/K7VdpceKbASYcr8Wd8ju2h6Suy8bam92k+GNr/w4vMpGXcoTFhx6I1GWNPUiEePkOnuAWBGXurSioWmM2XJvYRj59pZldqhbUb77q/S01Jz3Nc7wjDyDFUggt0ttO26C48nnAhT2tGI/tAcCufdj254UmZjUUYGfpcLE4mpLJr7+2l6T7GekWLCisPNf/4TAJP8LQih+H9nnZuWecoXfxxX3ucO22bY3uZo+8+o3/LnQV/jCMPocaoC0bTrVYKJcxmaUliW3TNDBG5m2pmpLY+f5PNn2mH2ZGjzc88/l5Z5TsSEFIddiRNuGe5efEaY0swssj3pOy+fW1xF/sz/AEDXLDzYWZVeYxWNG+8+IqvSEYbR52QFonbt93HJFwcOTSW3ETnT/52CiiVps89tGCwpLkaiBkKbr9fVpm2+YzEhxeErK15GIMn32enYn16Uvl9kEt3w2FmVoTlomp1VqSuF2xOysyq7WxxhGEMcTyCioT5ad9xJINCKplTi0JRGMFRiH5pKU5bl4VxRNQuAqDKRSnH/26sZ6bSDCScOf9ppn2zL9Xbi0uJcUjENXYzc25y27DYI3IwQCrcex6XsrMruxv/iuyteABxhGCsMJhCPvvoE3bX3AIltRKL4rGlcReWZd42YbUIIrp0z1w5tJpyTP35n9YjNDxNMHEwp+e3WzRhanByvfWz3/CnpKxp6LArLl5IzzT7gY+j2NuP/mueA6qa5XXeEYQxxuEAU9LZh+Xp5o6sYZSW3EYKMKV9lcpqPeA9GssZnHAtLSVbW1RGMxU7wqtQxocTh6397GYB8XxuakNyyMP3biWPh9mYmDm9NJmhBXCm6g366++y7gCMMY4ekQGRYLbSF/GztLeZ3LfMIRzIomHPfQLvB0eCOw5yTAJ95dnAndzqYMOLQ2tfHmprtxEN78Qvb1zBlDJRJK17yBd5qmstkI0Qw4iNHMxxhGIM8cd0NmK7JuKTAAHri+QTmfBmR5toKJyLP56PAH8BNH+XGOnyxrew4eGBE5p4w4vCjF3+Gx+imLKcZabbxqZwX2b/lKZSSJ35xGvnh26tpyJjG1shluD1V/OpT/+oIwxjll7fciTswk8xAFW5fDg+sXzvaJtGw46981P84lxStYVNfNo0Rwf+8+tCIzD1hStM3mRF8RpQMI8LCzDZ8mgn6Ktp2rSIcyqVw9h34Mka2tmJDj103oi+WScT0M7tkEgWB4R3QckgfPpd8d3BFAAAgAElEQVSLy2fO4qV91biFQUyZbGs5yIIRrp0RC/fSuO2XZAQO4gFQiv5+F9ICSxPsDA69ye+pMCHEoSMUoltUcXb+37hw8l6m0A+WwlIaFgKfv4u++nvpA6TrGoqrLhwRux7ZshmpNDrCtmPpe5ee3BFeh9HjjmVn89K+ajzCII7FM7t3Mb+waERWe621GyD0ewAyAiCQaAosKajU+xBSEDP95GUXs6utjTlpLI4EE2Rb8fj2bQC0qmn8tfUSYub7EEJgaAqPJnEr+5AMCrT4n2nd+UVq1v0UKx5Jm01vJQrZdkdyMaWLTy9anPbagA7DRwjBV86/ECEEHmHfO5/ZvStt80kzTs26h+z+JaHfo5RCUxIhJcrSBkKp0jwTl7cMd6AIhOA3mzemzaYkE2Ll8NK+agQSvysEwNT5VwJX0tvRQE/9A7g9EdyAUnJgNZERqKVj79dRgCvnH8ibnLomNnbYqZa4NOiK2FuZ6+bOS9n4DunFDn+/iQudmLDY3trCh2ZUpayxMEBPax39Bx7A5YqT4T9ylSAT92xL6ngn3UZuyUwAAm+vJhiLIZViV3t7ymw5FhPmVhZwBREozimbMvBYVv4Upiy5l4LZPyAYXjCwmnBrFq7EL0MAZvdDtO78InWbHk1Jn4rfbdkMQEe4AIXG999/2bDHdBhZfnnl1Qgh8CZWDz9Z8/awx1RSsn/L07TuuJNo209xueJoSLT3rBL6Q3Y6fsmC+waEAeyGxwBmwsku05wxOe5XDns77GKvGS67KOf5U6YedY2m6Uw741YA2uq3IvsfQRegC4mUYKFhAX7PBtre3UA0mkFe5R0Eck7dEdUTidDY20vY9A0kO6V7b+iQekoyM5malW13RhM6McviQF8vkzNPvclyuK+Tlnd/gd/fic8AkBiAlAKpDt2flfcTFE0/h2N1AF1YVMxzu3djSQmazsraGi6dlpqj4oMx7sXhkS2bgEPi4D9B+a+CqQuB+4lFgzRueZBARiMaEkMdcmB6PP0ED/yA4AGIi8spPclagAA/XfsOCkF7qAiA/7v2Y0N7Yw6jzg8/eAXX/fExvMKgX0ke3rSBb150yUm/vnnPW+jxpwDw+0FDggSpNJIlXELBAornfRZv4MTdIpNO0eTK4ZEtm8e3OAgh6oA+wALMk6lddypsbWnBECZePYLXOPm34/YEmHbWvwLQuGsFLl7AEAoDhVQCU4FE4OJFWne+SDA4mdIFnz3uoZvkKqYvmkXU8nJWaSlZaTwN6pBe3LrOdXPm8addO3ALnZgyWdfUyJmlZcd8jRmPUL/pITICteiAQmEoZdeHOWyVEJWXMGXBR07ZprKsLBp7e7GUpDuSPoc6jNzK4RKlVMo9KMk9VyCxanhf5bQhjVM251LgUruox74H8Pl6cQvbgWkqHQkEAgforrGrUIvApygoPzo1+4kd2xKhS7sH49cvHPl8fIfU8ulFS/jTrh0Doc2X9u1l2eTSo0KbnQfeJdrxELqm7DCkUoluV/ZqFCAW85I95bNkTjp663uyXDptOo9s3oQpJbquEYzFCLjdw3qPx2JcOyST4cJMVz8Ai4uHl6wSyCmi4oxvUTDnPsLRsxBC4NIkHk1iKIlQ2M1Yg7+jdecXqV3/EEraC8RX9lUD0BnJw1Iu7jjzLDQnE3JC8O3ll9qhzcS99LHtWwHbwVi78TFad9yJ2fUrdM0OQ+pSoiwx0M07GJ5Pwez7KFvyvWEJAxxqgmMp23H+1K6dwxrveIzEykEBLwshFPBLpdSDhz8phLgduB1g6tRT+4/7381H+htSdTRbExoVS64HrqezeTfRjocwNImBRCkwlcBCEPDvpO3dLxOOelndtRwpMuiO5AHwoRlVKbHFYfRZXFwCgFsziEuLA11N1G18Cr+nn4CdwoimVKKDd+JvUIDIGHyFmQpMpVBK8cedO/jUosVpmWMkVg7nK6WWApcD/ySEuOjwJ4fTDq89FMKjRTG0OJMzM1No8iHySmZRMv8+8qvuJRicjhDg0hQeYWEou0LQy31lWFYL+1pMQOPHH7oiLbY4jB7/e9W1AJSoWvxaPY0uA03ZK0oskFJDIegPlpIz/dsUzvth2oRhYVExKLDSHMpMuzgopQ4k/m0F/gwc3U10CCRLemW47S3FUP0NJ4vh8jLtzDsomns/0n3dERmYmg7huAcTi4ChMy33xJ5nh/FFvt/PgsIiuuOS9pCfl5sqiFpGYusgiIvLKZz3Q6addWfaK0VdXFEBHIpapKsAbVrFQQgREEJkJr8HLgO2p2LsZ3e/C0DAsLcUlTkj94EsmXEuRXPvJ7v8m3SF86kI9BCKeck24vzsiitHzA6HkeXfL34fPVE/ulIIJdjUVUZg8pcpnPdDSueO3LmZbK8dAbMS4vDotqE3VDoe6fY5FAF/Tnh2DeD3Sqm/pmLgR7dtBdSAv2E08AZyeYMP0NIfxNTKiRpe8gLp2d44jD6GrpObWYElLZQVZ7OcwodzR6/buSUlSilW1dfz5fNTP35aVw5KqRql1KLE1zyl1L2pHN9vhNGEZH7hsXLK0k9LKEhcGkQJUJKVPWp2OIwMZ5dNQdeNEa1LOhjLK45MpU4H4zKU2dJv+xmSq4aLyytHxQ5T2r+Y/ri9x7xl0dJRscNh5PjUokUA6IkTtr3R6KjYcVZZKXBIHDYfbE75HONSHH6f2GNlJPIb8ny+UbFjU6JcV3/cLuBy7mGHvhwmJlMTpQeNxMrhzf11o2KHW7c9AlbioOBvEmH9VDIuxWFlXR0aFn4jPKp2rKi1W94lxcEp/3b6oAsNBGxqTv0d+2TJ8XqRys4UrunqSvn441IcAAKuEKC4cOrIl55PEpeSiOnFUgZLE4kyDhOfj86egxACfZRvBpckwvdmIlsybg2/3MDhjDtx2N7aAhzaUpw7ZXSW8sG43T+g37RXDSPRVcthbJAs3GMIuz3eaDW6nZtIGkz6vl6p2ZfS8cedODyy2S6kknRGevTROXW+KnGuI7mlmJabNyp2OIw8WR67q3ay7N/K2uN3VE8XyW1sMlPykRT7HcadOOzuaMcQcTx6lMw0nUY7GdYeaEIpQTA+MpWAHcYWAZcLXWgIIajr7h41Oypzc1FKYSmLsGme+AWnwLgSB0tKzGgII1JNLNTOEm+MWHT0kqBCpg+FxpVVM098scOE4pbF9jYyme8w0k1uAcxYhGXuXhb4t3NOxlsUaM30RlNX42FcVYJas3szVmw/2dntaFqEWa7n6Nr33BHXKCUIhYrQPdPILJhPTtEMNE1PqR0H+/uAQ/kN189bkNLxHcY+7582nZ+vW4uhaZjSYmdbK/MKi1I6h5KSntZaelq2Y0Wr8XpaMIwjVwe5wOJJOtmuKFdnbuONzXP48NmpSeEfV+JQ4PWABgFDUhXow4WGLdgKhUICQigCgYPAQeJdq2l7T4QnGvVhWmV4MmaSV7ZoSI1uVtbVAtCX8DfkeEcnz8Jh9Ej6G5L5DitqaoYkDpFgNx0NW4j27sHQG/B6+4+6xmcw8EkViRWKQCEQoBSemCAqDIQG+/vahvaGBmFciUNV5TwK/D4uKm5kYW4brsjRuyKpbKFItKlIyAYk6vXi8YTxsBfkXnrrn+e9fuZwKB/0cjLy55FXOhdNP9qvUd3ViSV1IpYXt9OL4rSlKi+PvZ2daEIcM1NSWiadB3bT374dGa8lI9B61DUuwJVshKYUguR+3xYApewVsUpIgv3MIbQoSI/gwd3L2W/q/FOK3t+4EgeAXlHB2h7BjmCEb160nFg0SEfDFsLdu9FEPX5fL3aljcFerSOVQqpDogEgE9KhhMLn7wA6sPo30rb7yFebpototIR4uJygmgwIblnspEyfrnx60RLuXrkCn2aSrzdSvfaneFzNeNxH7/v9bsANSoEmFNrAp/uQAEglUGgo7IKrgxGNZGLKMnzZs8grW4jHb5/nuffN18nwptZBP+7E4b24PQFKZpwHnHfUcwpFX0cT3c1bMEPVeDzNdq8AAUeqh63TEoWSoITtYEoKSPJfw4jTGO9GkEdXbzsxK5cPTncqPp2uLCoupkBr5mPlr9Ic9WPEFa64faPRSJQVtBf/oOzVq0oIwLE+/NIyiESK0b3TyS5eSOakcjRtdJKtxr04HA+BICu/jKz8wasFW2aUzqZd9HfsBKuOQKDjUPxmkOw3qRQBAZrUCZkGMtqJW0+ts9NhfLEw0ERRoI/WqA+BRFj2WkBy7L+LcDgXJcrx580hv2w+hmtsViif0OJwInTDQ0H5YgrKB6/BF+prp7NxC7H+PbiMJjyeMCWeMPMyOmjryCBkjN5ZfoexwXVnfJ6/7f4SlhJITcPAIh7zEotPxp1RRW7JIgK5qY1ijBSntTicCH/mJPyJsvWHU/fmSvqyK4lYo5eE5TA2KC2Zzh9WXkt/xE6n/+bFE6cdgeNqd3BwGBRHHBwcHAYl7eIghPiQEGK3EKJaCPHVdM/n4OCQGtJdfVoHfobds2IucKMQYm4653RwcEgN6V45nAVUJwrNxoDHgavTPKeDg0MKSLc4lAINh/3cmHjMwcFhjJNucRgsteuIs61CiNuFEOuFEOvb2lJ3aMTBwWF4pFscGoHD67iVAQcOv2A4vTIdHBzSR7rFYR1QJYSoFEK4gRuAZ9M8p4ODQwpIa4akUsoUQnweeAnQgYeVUjvSOaeDg0NqSHv6tFLqBeCFdM/j4OCQWpwMSQcHh0FxxMHBwWFQnFOZJyAW7aezYQuhnncxqMfj6yceWUxPj4bmmTra5jmMMj097QSf3UisxS4T9/s3NlBWNYnSmdOYPHMBvozx23n9tBcHJS26W/bS27odGavB7285qs6LBmQkasiaCEJhHUOL0h06OOL2OowtHvj5DxBr21BlmQhd0NMeo6f9ADvePgCsOur6ydMyKZtZQunM2RRUzEIfw8WCTgtxCPW109W0hWj/Hly6XbTlvfhc2JU+gVgUYlGNaEQjGtKIhAWRkE4kqBE2dbRgkHOmhdhszEYqheY00D1tqfX7yMqW6IaFEAJhgkCiSBSIEwqEsKuUaoIDNX0cqOmDv+45aix/pk5pVR5lMysonb2AzNzCkX47RzAhxMEu97aTYOdOsPbj93ccdY0LcPmT10M0KohFNCJhjWhYIxLSiIQEkaCOksf+sAsJ+YbJjgPziE+dzlv1+7mwvCI9b8xhTFPb3cWBoiq8t2VStHUf1kGLeBTUQO1oAAEqkSps2o+KZHFjIeyrhEAJQajPYu/GNvZubMNOETqSgjIfZVXFlM6aQfH0ebjc6S0vN+7EIRIJEjUacHuitO6884jn/HYLQ5RKfviFvQIIJT/8GuGQhhk9vh82HM5GMhVfzmzyyxbg8gaOeP6hN14Dl44O/GbzJkccTlN+t8XuTdnpzUMuy+WbFy4/4nnTjHKwejtNe/bSuLeF1oYwHFZe3iYhHgoY6ElhV0JHHVp5KCFoawzT1ljLppW1wCtHzBU1fbg+MB1P6an3YTkW40ocmpprIbYfhEU8BsE+QSyqEwmLAQEIh3SiIWH/xx6DeNxFNDYZwzeD3MkLyTxGAdpjUZKRSXN/HyoSp43QcN+WwzhlffMBlFJIqQaa2xyOYXgom30GZbPP4OxBXt/X1ULznu007K6jaV8XwR67m9WARCSrpCfEQyiV2K7YVylhi0e/aWCGLCI/3QT/ckbK3t+4Eof6lgY+lF9Nb61FpFOwLTvzmNcGQ/mgV5KRN4e8srnoRurqPb6vspJHt21F9UYRXhfheByfy5Wy8R3GPsnemJZll6J/X+W0Ux4jM7eIzLOLmDmIckhp0V6/h8bdu2ja20xjdZ8tBhwmHgBK0BdxYcRMDCGZn5m6xs7jShyWzj+PJ995gcXGJmJanLaeCnJy55Fbtgh/ZuqWUyeiMjfP/qYvBoXw7O53uX6+0y/zdOKdRrsSgWnZH9WlJZNTOr6m6RRWzKGwYg5LP3j085FgL817t7F35z4ObuzFHRB43z+bL13/iZTZMK7EwWMY7MmZz27vbOSBdooyMrhtzrJRs0f1xVFK8ei2rY44nGY8stX2NyRXDsYIt0X0BrKoXHw+W3x5uMpa8fkMdF2nOOPYq+lTZdxlSBYHMhBeA9waLf1HNx0dKeYWFIKUEIyNmg0Oo8eBvj6kVCipKAgETvyCNLG9pRUEaGkQp3EnDrcsXgKAyLJDE5aUo2LH8opKAFSfLQ6jKVQOI0vUtB2HyVXDpRXTR9McdF0ghODGFK9ex504nFdmpyyLLNvBuKm5eVTsyPPZKZOqN5E2u23LqNjhMPI8v9fusJwUh+l5eaNix/7ubgB03f4Yf3T2nJSOP+7EQSSzETPtlcOKun2jaA0QNFGWZGVd3eja4TBi/GbLJpRSWNYRcYMRZ0VdDQCGbluQ6ojZuBMHgKXFJQhDA79B3BqdbQXABVPLAQX90VGzwWF0kFKhlGJW/qRRs+FAbx9CCDRNJz+xkk0l41IcPr0o4XfItLcWodjoOAXPKbPLY6pee/4dra2jYofDyNEespPekluKpO9ppJGJPAvdsFcNSV9cKhmX4jAtkWeQdEquaqgfFTu8hh0JTvodHkmk0zpMXJ7YsRVgYEtR4B+dSMWWFvtEsK7Z4nBRGlL4x6U4DJDhBg3WNjWOngkuN0QtVMzi3fb2UbPDYWR4aV91wt8wettZgL/V2v6G5JHvdJwMTlsSlBDiHuA2INmM4uuJepIp4cqqmTy/dw8E3Ham4iigpOICz34KSjbiz9Wp1iYTi38Ct5NKPWE5x3yVuaUH6enz0tRShZIXIUY4AQogHDfRNIGmCRYUpudod7rf1Y+UUosTXyktMnv9PDummwxptgRHLs8g1NfO/vXfoW3XXUzxrAcURjjG+6fs5Z4VL42YHQ4jy682rOPSOfswdIVyCSon7aXt3S/RsPFu+rpGLqQeTuRZJEOYSR9cqhlX6dOHk+O1vbMiy4Nq6mdlbQ03zF+Y1jmb97yBHn8GAJ8PhG4noOimRl+7izc2F7HN1UVnODyQB+EwMYiaJs/t3U2rdyZl7i5QYEgLqYHHGyLc/F+EmyGqPsCUeR9Kqy2rG/YDh8Rh1qT0REzSvXL4vBBiqxDiYSFEbqoHd2sa+AwwNKo7O1M9PABmLELtup/QuuPOAWHQXALNrYPmQioDtxS8vXsGf61fjhAatz79VFpscRg9Pv/icwC83nEBK5oX4YkqpNLA0tClQkcCCo94hdadd1K77j5i4b602PJ2QyMI+8aUToa1chBCvAoUD/LUN4BfAN/BPmH6HeB+4O8HGeN24HaAqVNPrWDrLYuX8quN6xGZblRXBKXUoSSpYdLRtBOr59egIOAHoSs0XUMq3S7WoSAa8ZFT8Vky88uwrC3Q3YXqiiByvWw52Myi4pKU2OIwujT09NASDGLGJZYp6XZNo3j2zYT62mnf8wu8vm6wBBoKoUmkEAQCB+muvccewPdJCitTV2cB7CiFEIIPTp+R0nEPRyTPpacTIUQF8Bel1PzjXbds2TK1fv36kx43Zllc98fHkO0h1P5erps7j9mTht5vU0mL/Zsfw+85FJLUDEAXSKnbNQGB/vBCKpfejNAOFQdVSnHvm6+DoSFm5iB0jWdvvGnItjiMHa56/P9QShHpjyOl4svnXYhbP/J3X7/1SXyutwce04QEASbaQMHi/uAcKpbeiqYP/Z7cHgrxwPp1uN06brfBw1dfwyT/qdVwEEJsUEqd8DhzOqMVJUqppJfmGmB7qudI/oJEphsF/K2uZkjiEOxqprP253g8Ifweu7Sf5hIoNJTSwP49o2XeyqSpCxjMNyyE4LJpM3i5phrawlAc4NGtW7hp4aJhvUeH0WVlIkXZjEqkVCwtnnyEMID9uy9fdB1wHT0t+wi1Pgi6CQp0FEJXWGhkBHbRvvsrxOMuAiWfJaew4pTtea0uGcK0PQKnKgynQjodkj8QQizG3lbUAf+YjknKs7PZ39MDXp3O0NFVpY9H486XcKuXAfB47FWC0DWk1JCJMnP9wTKmLLodl+fEyS5nlZXxck01qj0CeV6e2LGdj8+bf9Qfk8P4QCnFj95ZjZKKeMwC4IoZM4/7muyi6WQXfR/LjFG/+WEC/r0oSyAUaJpCaeByxYm1/5TWdgibFzB1wUdPejv8bnuHffPS0n+iI23ioJT6VLrGPpxbFi3l22+stFcPkTBxy8J1nA9jLNJH07YHCPgPYgdBFbpbgNCRUkMlclvi2ocpnXPJoKuE4/H3i5fy8OaNqOYgojyLu17+Kz+5/MohvjuH0eSH77wFQCxqoZTiujnzTvqUlW64qVz2WQBaatYgIn+wbziWvZpAU1hCw2esom3XKkLBPIrmfA5fxrFPeCYdALquIYTg1jSkTB/OuA1lJjkjUZ5LZHpQbWHWH2ji3ClHOzZb6zZC8FEg6WAEYQiUMuxfmoJwOIuCGZ/DnzP0pJLJWVm4NI14bwzVH6OObg7296W0Qo9D+umPxXh9fx3SkpiJVcNQ/VlF084GziYa6qF55y/w+9tACjQFmi6RgD/QSV/9vfQBpusaJlddcNQ4+xIRueSW4sqq469ihsv4Tp/m8CPcbhCwIpFWCiAtk9r1D9G6484BYdBcoHk00FwoaYAShGLnUDDnPsqXfWtYwpDkC2efC4BqDqGU4vbnnh32mA4jy98/a4ejYxFbGP7pzMHqR58aHn82Fcu+SuHc+4la70MJsKSGkoeFQxUY8T/TuvNOatb+D5Z56MTvilq7PEFSHDxGeu/t437lAHBu2RTebmwAvwuCcXrb6uhregCXK07AB2ig6aCEPuBgNC0NX8Ht5JZUpdwen8vFvIJCdrS1QmcE8n2sbqjnvEFWNA5jj70dHURMEzNuYZmSQn+AXG9qk9qmLLgSuJK+jkZ6Gn6B2x0ZCIdqmsJCkJFRS8eeryOVwJ17G23BECKRMj05M/0r0REJZZ4spxrKTNLY28MdLzxHedsmzmcjnrBEkwkHo6bZySrJMGRwBuVL/h7d5Umx9UcyENrUNcQsO7T5zA2fTFkehkP6SIYuw/1xlFR89fyL0l5AVklJ3abfEfBtTTyQcGAKkEKgBHS5PWztquSAtoCvX3jRQMmAU+VkQ5njflsBUJaVTbHYxycXvoMClNfOYFTChVQ6IJCej1M4736mnfW5tAsD2Nudq2bNBkuiWuwaAA+sP7rFmcPY4rk97wIQj1ooqTh/SvmIVJYWmkblGbdQOPd+tMxbQIBUAiUFulTURzN4cv9M9oRc5Ki9nF16ao2YhsKEEAeAWa5e/NkSd0DDsuzVQjA0iayKuymcdz/FM84ZcZsWFiWSRzsiqIjJi9V7CcfjI26Hw8khleJXG9cjpSIetX0Nl4xCMZdJUxZSOPd+cqd/m2CwDEsKnm+uIh430IQik+iIrEAnjDh88Ox/ZsXOaQR1H2/XVvFM5AYqz/wa3kDOqNr12TPOJEP08lHrST5f/hu+/ef/HFV7HI7Nd576MXcU/4aPev5MfqiZm+aPbgKbyxOg8sx/ZbX+DyiyQIOY5ePqJTeMyPwTRhzml07lHddV/Ml9I3uKFtEaDNIVPrWkqHQQrH6Za0vfJBAP4w1HWOBfxz+/+Pxom+XwHn7w1psUBdaQ4YtheBUfKVsL+0Y/yhSOm+xoa8EbyCY7fxo52eVcNmdkRGvCiAPAw1dfA4AosbMZf7ZuzajZoqSkYcPd+IxVeGMWuiY4WJvB+r0V1HZ3OwIxhvjBW2+yqmE/6ztmUdeWj67AAwT8W2nZcRfSMkfNtp+sXQ2A22cghODBD189YnNPKHHIcLu5uLwC4TMgzwvArra2E7wq9YR6Wmnb9SU83hCaoUDTcUcFe/ZMYk/3bFRf3BGIMUJSGKy4ZH/vVNa0zsAVBqSGgUQIRfvur9Dbtn/EbTvQ10dcSnSXhm5oVOTkjGgy3YQSB4AvnnseAKLID5rgyV07GMlw7YHdf6O/8fuAQnOBFC5AQ/hu4qyL/wWUQjWFHIEYAxwuDLH+OErBBXNuxpN9OwCWpeFSdmJSpO0n1G95ekTte3jzBgDcXjsd6b8+cPmIzj/hxEEIwb+ecy7C0BBFduLKC3v3jMjcNWvvwzCfB6EQbg2ZyDHLqfwWhZVnkOXx8oWzz3EEYgwwmDDctmQZxRkZ5JXMJL/qewCYUkNXCoHC63qT+g3fGpGbTbJossujo2mCT8wd+QN8E04cAC6pnGZ/k+8Dj86mg83EzPTtG2PhPlp33ElG4CBCV2DoKKUT7C+mcO79uH1ZA9c6AjH6HE8YkhguD0Vz7ycYmoVUAmGBriReXz9tu+4i1Ju+7apS8HJNNUIIXB5bEG5euDht8x2LCSkOAD+74sMIIQackz9ftzYt87TVbaC75h4ANDcoYQAace1KKs/60qCvcQRi9DgZYTicactuB98nUQik1DAS24z+xv/kwLuvpcXG32+3+666vTpCCL6z/NK0zHMiJqw4TMnOpigQsLtiZbrpj8doTXGF6pp1D6KCv09sIwRSGSAEGaVf+f/tnXl8VPW5/9/POWdmMjPZWEKQLJAgIqAsClGsoqLeLrYItv0Vqy1dvPdXb9vbXavltretrbbV21/vr7eL3rbaXndtK5ZbbasVFUEB2fclCyGBhCxkn+08948zYR0gmSWEcN6vV15JZuZ85znJOZ/5fr/PRtGFc095rCsQA09/haGXwrJLyS/7N8BJlLLURlAs+wUq334grTZ2hMNUtrZgmAaW15k1nKlyg0NWHAB++r73A3HXpggPrel/3kYiYtGQs4wIbMcwFYkvI3p6AhRM+hGBvL5ldroCMXAkKwy9+Pw5FE5+kK7OQmJqYMTAVCWYXU/Dlq8Q7knPB8//eyvuusxyhOHRmz6YlnGTYUiLg8+ymDdxIuIzD7s219Wn1l+gpX4nTdvvAYh7IywUg57IlZRe8l1E+vcndXJPM2MAAB0ASURBVAUi86QqDEdTNutOosb7sRHsmDjLDKB1z7dorE6tHWJlaysAVtx1Oa1wNMPOYIuDIS0OALdf4iSfSaEfLIM/7dye9G5z9bqniDT/AkQxvOJ4I0TwFXyO0mkLkrbRFYjMkU5h6KXowmvJLroT4vUYLNvZh9DO/2bPqv9KetzHNq4DwBN3XX77DO019DLkxQFg8ZyrEdNwYh+A57Zu7tfxasfYt+5u/J63HW+EJ16iXmHkxPvJK0g9Oed4gcjt2Mt4fYJvPP19IrFYyuOfa9iqfPnJX7J9/bPkHNyTNmHoJZhXSMGkBwiHgs4yQ51iLdnBrTRs+QqxaP9aNC6rrgKOuC4/PePSjPS/7A8piYOIfFhENouILSIzj3vubhHZJSLbReTdqZmZGhW96a3DfJBlse3gQbr6mB3Z0VJP49Y78XjCGB51vBFq0tl1EYVTHsQw09cXs1cgRpn1zDOWMcNfwwfK/swHH/8dT23amLb3Ger8eecO5j/5OOtbaqkih7ZoN+Pbt6dNGHoxRCiZ8R16IleiKtgxJ6oSoGnH3bTs39WncaK2zes1VYhxxHV508RJabMzWVKdOWwCbgZeO/pBEZkMLASmAO8BfiYiZ7QE86/mORV+ZYzj2vzxiuWnPaZ2y0t01T2AohhesPGACEb2IspmfjIjdvpiPdw0bBUK1FXn0lgVwLYNHtu4gXlPPMamhgMZed+hwJ6WFuY98Rg/X70KtZVYp0mszSLf6uHa0h0Ewocy8r5jpy3AV/A5wImqdNydSqT551Stfea0x/9m3TvAEdflgwMcCXkyUhIHVd2qqtsTPHUT8KSqhlS1EtgFVKTyXqlSEAwyYfhwJOiBPB+K08noZFSt+j5e/Qsiiuk1HDclMGz8dxhZmrmenIeqvosoBMKK0RXjldUXQk032tSD2so9L/+NeU88RmvPmc84HSx0RSLc+twzfPHF/0FViXZFCDWFEXs4BvDe/GoMoHv/v2PbdkZsyC8oo2DiD4Bed6dioAR8K6ldew96kvdt7enhQGcHpmlgeUx8psmEESMyYmN/ydSeQxGw96jfa+OPnYCI/JOIrBaR1Y0ZTpL6wQ3O6kZGB0CER9efuLvc09lKw+avEAg0IZaCaWKrSWdHMaMmP9in/hXJUrXqPgBn+aImdlc+V7zro84+RHMEre5C25y17Mf/8Hu+/eor2IOozN9Ao6r8x1srWPjs07SHw8RCUcLNEaIdzgbhxy+dzaLhLXjFjveyhPoN38iYPYZpOe7O7ouIqQFRwcLG6wvRuO1rdCToxP3TVSsB8PqdifWv592cMfv6y2nFQUT+JiKbEnydKnc00U5KwqtYVR9S1ZmqOrOgIPlWdn3BMgxuvXgq4jWRAsdFtGJvzeHnD+x+i7aq7wLxGxQLFYOoZwFlFV/KqG21W/5KIHgQw1RitjNLKb30m5Tk5rF4zjVcVz4eojZ6IIS9twvtibGmvp75Tz7O0h2JJm9Dm9erq7jpycf525492NEY4dYwkUMxNKbMLi5m8dVXM37EcMoq/gXA2Q9QG8sbpvKdJzNqW9mln8TMWeRUl44diarsqn+A2i1/Pfy6bQedD0PLa2KYBleVjiXHl/kShn3ltOKgqter6kUJvp4/xWG1wNHVL4uBulSNTQcfuehi54cCx7X5cuUebNtmz6qfIj1PH+WmdPYXckruSdhDIJ10ttTj5UVEFBUTEcguufuYUmCzi0u456qrKc3Ph54YurcLe383GrX55ZrVzHviMXY1N2XUzsHAvrY25j3xGD96c7nTiao9Qrglih1Whvv93HXllVw3fvwxxww//14A7Hh150DWKprrMiuoI0umMqz8O8BRywwFLy9S+fb9oPDs1s2ICN74JuRXZ2f2OusvmVpWLAEWiohPRMqACUBmkhuS4PvXXY8YTt6FQYx9lXeRHag8HO1oq0l3Vx4Fk36EPyez6z+1Y3TWP4DTsMBAEaLWfAI5I094rSHCx6dOP9wXg/aos9RoDqGqfPmlF1n47FN0hvvnRjsbCMdi3PGnJdyx9IX4vkKUUHOYWLfzqXzHrFn8c0VFwm5nls+PmfcpQDBiAEqk9SGikdAJr00n3qygk7zVUUJMBWwwsQlmN7Jv99fwEMbjMxFD+OJlVwy6yuSpujIXiEgtMBtYKiIvAajqZuBpYAvwIvBZVR00zvqLRhUCYObB7VOWYpsgR0c72nMZO/Ob/Y52TIZ9G/7VscUDqgYdncWMueCqUx6T6/OxeM41fPTiqWAr2hR2RKIjQlckyi3PPcOPVywf0DoWmeSRde/woaefZF97O7FwjHBLmGhHDGxYMGkSi6++mhGnaSg7ongKnT3TsFWw4n+Xpp33DIT5lFd8EdtagAJ21MBWWNVWyPjAJrIsZ2N5bm8m8SBiSPStSIbWnm5+sPTrzB+9mprtQegErwqBwi+RMyK5fgD9pXr9M/g9KzEtJRav/TBq8oP9HueVqj28WRPfOwmYXFC0jQUT17KqoYRXW2/g3eMnsHDKxae9gQYLbaEentm8mee3b2NWzmtcU7STP++ZzDuNl2KHnOt1amEh8yZOhH5+2h7Y9FVEFMOyiWLQ2TGB8orPZOI0TqCrvYnm6vt5uG4atqlY/hhjc8J8+8a7Kc0buELIfe1bMSQ6XiVDfpaf87MvwhNYxZiyHmreCfBo6/v5wvjU2+H1hUMHduP3rEREieFMhfPHfyepseaOK2dO6Th+tXYNHd0NVER20NVgUpFXzStNNi/t3slLu3cCzk7xomkzuHHCxIy3U+srUdvmL7t38ei6tXQfVXdDVZmVX0X7IQ+TPHVURYvp8JzH52ddjs+TnO0Fk+7n4La7sKOCYdkEs3dyoHI1hWWnvVdSxgzk85vmG7HtejyBMIallAeKBlQY+sM5O3Po5Y7Hf0lD2CYc8qN1naDwhctmZ3TX2I6GOLjjHkARj4GqgWR/jILS1At6VK9bDPlh8IGtBv9d8z7wGYjfRPwmeOWYte2YnBw+Me0SLisqHtA17/r99Tyybi27W1qOedyOxNCQjR220TB8rGQpmDYxMcjqthk9KfUU6bbGanoa/gMRxTacblL5476FL5B7+oOTJBSL8cPlbwDgDVhg2cRiEV5cdPuA7zX0deZwzosDwD8u+SMHOjvRtsiACETDlq8Azj5DTE06uqdQfumnUh5X7RiN2+5EDCVmmByM5vOHQ8ftXxiCZBngNxG/gXiO3VeZOaaIj0+dzrj8YSnbczR17e08tmE9r9ccW6hVY4od6hUEBfvY6/GG7PWM9ddgGjYxDPLLv43Xn3oIdM36JWRZyzAMm2i8o1VhEku6vnC8MFjxdOwz1R7RFYd+MlACUbn6YYKBbSnvMyQc+51HCWZtAAsUg+ySewjEvS2NXZ38vaqSHU0Hjz3IEmdW0SsYpnOxRkOdFEkn9914CyNHJFdspK2tmWefv49V3Xnsy3GS09RWNBzDDtloSNHYsddfWf4wri0rP9woNhbpoWnHNw7fxB0d51F+2VeTsud4atb8G1lZ7RhmjKiYdHaMorzirrSM3ctgEwZwxSEpMi0QjdXvoJ2PIYaihtPDc+TE+9OWvOXMSBS1nAvwZKKjwM7mJv5euYfGrs5jn/QaqM8mYu8Fv80of4g2I7nlTltzFR6rm2sDVaw/NIt2ctHIsddbni+LuWVlTC4YddKbZf+mr2GIjVo2isGoKekRU7WVxq1xoTFtbDGIyvsompSeVOnBKAzgbkgmxcPz5jsCQScQROs6+clbK9IiEJFQhyMMomA4Xb+9Iz6bNmE4uNfJ2jQsiAFRa/5JXyvABcNHcMFwZ1YRtW3W1NfxSuUeYmGbSGsrhgi2xySQE6U53JOUTZ7uCCoWrflBJrdvYqXfic+4euw4LisqxtvHDVHfiNuJND+EpUpEoHbzXyie8g9J2XQ0YgjZJXfTsfc+JCYYJlj8D52tFxHML0xp7MEqDP3hnKjn0B8enjffqT2Z63EyOAV+8tYK2kOpBcy07P4WAGIJikF35F3kF6bPt223P+J8j0euny5W4mgsw+CyomLuvnIOi6+6ho/kV3JJzn6uLajmXQW1SdskIghgIEwY2cniOdeweM41XDV2XJ+FAWDYeRMBJ9IQFC8vJW3T8QRyRxI156EIEi/a0lH3w5RyVoaCMIA7c0hIumcQlaseIBgEw1JstQj1BBh7SfoSbCKhLgAnLwOTjs5xpOKQHRGso8CwsU2BkPCvc65Japx9G76OZUVQU0n1c6grXEHA+zYmNjGEjpZ6soelp/DqmAuvpvLt1QSDdVg48Q+16/6V0hn39nusoSIM4M4cTkq6ZhB1218lGKx3btx42nfxjOTiGU7G3g0PAWDHL77S6f+U0niGYTtrDxF6upP3DIRCY5zxUrLGYez0Dzs/xDOfm/f8Ig2jHqGswvEg9RZs8fm6qVr3XL/GGErCAK44nJJUBaK7rREr9gKg2PGEquCYu9J+oWQH94IothooTkOWlBFnWh3V5KNFLb+zbErH6YoYdPfkoLaBiZKV1YGmuXzesHgQmh0VDFX83jdpPbCnT8cONWEAVxxOS7ICoWrTXns/Ts9MZwMyLDcSzE9vBGbd9lcAEFMQASv302kZt/d69udOTHqM/POcVvESz9Y/WcGTvlJw/h3OePH9gKp1T6U03vF4soJI8DbAEQdRCDX9J7HoqUsKDkVhAFcc+kQyArF37bcBMCzBVoOOjkKKJ5260U0yWLGlKHp4I3JE8eSUxouGnUSg3mt6REnyUZs5I5zanb17e619rKl4MgJ5jgfBjgmgBLPWpDReIgrGzaCzexK2bWDGy84f3PH1k75+qAoDuOLQZ/ojEDUbnycrqwPDVOx43kR5xZ1pt6mrrQEA0wIQOrsvSXnMplrHJarx69rnT77lu/TW/ImLQ1vDplRMAyBi3AgIVnzQptqtKY95PGUzbwc40v4O2LP61ye8bigLA7ji0C/6IhBtjVVkma8hhrPPAJAXb6WWbhp3/hw4shE5bsYtKY/Z1eLcbJqwmFeSxJXGDu9Oeaii+OyrtwBAtPVXKY+ZiJET7wecDUpTbYKBzTRWrz/8/FAXBnDFod+cSiDsWISexv8Pqojp7DOo/5aUPn1Phqri97eBoaga9PQEESP1f6dBTXz89F3gvekSfn96Kmd3dhWjamBiI6JEkwzSOhWG5cE34p+dX2xnn8Pu/C3hns5zQhjAFYekOJlANGy/G3C6bdtq0NF5QcZSgWvWOyXPjXjho+Fld6RlXL/facmmKmkRiJ7uXEBAbAwjPaH6xVMdV23vxmTN2uS7TJ2KvNHj6Y7MdoQo/l4te755TggDuOKQNMcLxNTARkIBQeKBTgDls/5vxt7f730LRONRg6QtIAhwQrwFujpT96zY8VKi6bzQeiuA2zEDQcnOrkzj6McydvqHCIezsG0DbGWvkU1FzqohLwzgikNKHBEIkxsv24KYSrfHJGrbDJ/wvYy9b8t+p3CLYQEihDTNDcXi17l4Um/zl5U/KT5kehP8jJxF9LocAeq3v57W8Y+maOp3WN5axM9qLmHJ3vPZFMrH43M2KoeqMECG2uGJyDgR6RaRdfGv9IazDSIenjef0X4Lb55NyYQOfAGb39dfwf0rV7L+wP6MvGfo4MMA2PF/X0kakpCOprd0Zm7BlJTHGlHsNADqdWeGezpP8eq+09tYSG3HrWlG/5iWcY9nZ1MT33vjDTaGpiKmYgVihEwTL11DWhggQ+3w4uxW1enxr4Ep0neGeGj+Layv+zAr95byVPd1HCotBlN4Ycc27n391RPTolMgFg1hGDEMU1EVOjvTt5zoaD22e8CwMckHQPXijS8Bem+h5toNKY/ZS2fPxaCCGZ+VdLWlrzR/W08P9y5bxlObNoGANy8L78giIgSoyB3F07fcMaSFAVJMvFLVrcCQ/yP1hdvf/wUOdHTw9z/90bkRsi20OQyHIvxyzSpyvD7umDULr5Farlv1ul+THQCN/82LLk6f7rbs24BP4mMrGMaJZd6TRgUEulu34hQrT51xMz5G49Y7nTgKgYZtP2dcxeKUxozZyqPr11LX1g4CZpaBFbQQQ7Cw+MVNn6FsWHqrZA1WMpmVWSYia4E2YLGqZm5ROEgozM5mycLbWLG3hvuWv4YU+NA8D9oYor0rxA+Xv8HMMWN4z/gLkn6P7MAupyitOrka3qz0dY2Odu3EFwS1SdyzLAXs+A1sHNMlMTXEMAlHfHgJYVg2gWALaitiJGf8GzXVvFpZBYDhEaxsKx76Dp+dVcG7z5+QLtPPCjLVDq8eKFXVGcCXgcdFJGH1zoHslTlQzC4p5fmP3Mr7zr8A8RoYRX5kTBZ4DVbX1XHv66+eWK6tD+zf7fRVFDMe3hy4La12e71OL0c7zf06VCUeCBWPzUgjeSWOC9eILy1qNvyh32PUtLZy77JlvFpZhZjgybXwDvNieAyuKCnhjws/es4JA/Rh5qCq1/d3UFUNAaH4z2tEZDdwAXBCDThVfQh4CJwycf19r8GKiPCZmRV8YvoM7lj6Ak10QcCE1gjaHOHpLU4o8edmXU5+VlafxjRCzzh5FGqAwKhxM9Jqs8cTojfWORTqm019oaurkGBwP4KmN/ISyBlRQvd+p1mMmIrfsxxnG6wPdoUj/PuKN51fBKyAiRkwEREMgd8u+BC5g6h35UCTkWWFiBQAzaoaE5FynHZ4fct9HWJkWR5+c9PN1Bxq5XN//hMM80KuB20KwaEoP121ktHZ2Xxy2iWYp4hw7Ok6BDh5FDGEjs7JKRV0ORm9k4ZINGFT9KQwfeOB/RiixDIg/yG9Fp/8HUOcQjCt+/eQP/rkVbZUlac2b2JXU7NjX+++Qry47g+u/wcmZbip89lARtrhAXOADSKyHngW+IyqNqdm6tlNaV4+Sxbexr9UXI6YgjEqCyn1g99kf0cH9y1/jddrqk56fP3mnwFHNiLHXbooI3b27i37gumbRuf0ukTjwpDuosYlF73f+cF2jO868NBJX7umro7vvfYau5qaEUvwDvPgyfUgpvCJ6TNYcsutrjDESdVb8QfghEWeqj4H9K+MzjnC9eXnc13ZeH745uss31uDFPvR9ih6MMSy6iqWVVfxsanTGXtcF6Rg8KCTzKUm4bAPI0Wvx/HYdrzTVLzIy7CiaWkbO7/wfBpbjuxxdjTvO5zOnS66ugsI+BsxDBvLimBHIxjWkeK9Bzo6eHiNk+ItBlhBE9Pv/A0nFxTwvbnXn3Lmdi7i1pA8A4gId71rDu2hEJ94/jkiORZkm9ASQZvD/G7DOgC+dPkVBD1eajYuIct0NiJRyBmTWhm4RLTs29prHCgE89K3aDnsEo17LFrr1qddHEZP+gxtVd91tE2g+p1HKKv4R0LRGD99eyXdkajjmvSbWAHzsEfj1zctYORZ0kN0oHHF4QyS4/Px3P/5KNsONnLn316C4V7ItdCDYWiP8uOVbzJ+2DDmmssAiKnzyZY3alzabWk/uJmAL82p2sfR686MdKdW9CURWcF82gCNOfktweA2Xti2jfUHnExQw+fsKxiWc37fuvpaLh0zJu12DCXcedQg4MKRBSxZeBufmn4JYhkYo7OQYj9kGexvraEnaBANODdud3ROZoyIVgG94cgZIp7l6Yu7TNM+vP8j2Ag13dnUh7zs37MGsQRPvgdvngfDEm6eNIklt9zqCkMfcGcOg4j5F07mAxdcyDdffZmNDQeQkgAzZQWGKhE1UcvgdzW5jFqzmrll5YwfPjxtn/OBoBNjYmdo5hAOBfD6ugDF4zl1Tcb+UnOolVcq91Db1kW0ezoqSsyCy/JqaB1WjogwJieHn7znfYOms/jZgPuXGmSYhsH35t5AU1cXn1zyezr8fsaPa6aj3cfyXeeDZdLQ2cmTmzYePubCkSO5tqyMEf4U185CPFV7eGrjJCASK8LLTkQ05ToRh3q6ebW6io0HjiseY0LMm4WZ1YHltWmyg4gIP7/xAxTlZq6D9lDFFYdByohAwAnFrrqMNzY/yP6wl42emZglJhqx0e4YdNtoT4xtBw+y7eCRiMsrSkqYXVyK39O/f69hqNMWwhib3pMBvNkXgL0TJ06yf4SjUd7aV8uy6qpjnzBwIlC9guFzOoZ7KeZQaz1FtPHBd32c9069NE1ncO7hisMgZ/a4Ccwe9wtUlRW1e/nt+rXUtXcgHgNy4zEDIRvtiQtGyObNvXt5c6+Tw+C3LOaWlzOtcDTGaRPknNs2MDy1CtaJGFE8jbaapYerN8WiYUzLm9gKVbY0NvBy5R7aji7gKyCeuBD4DMQjxyT9Dc/ys2j6DK4eW9aHc3U5Ha44nCWICFeUlHJFSSkAoWiUpTt38Mi6tZBlIlkm5HtQWyEuFNoTozscZemOHSzdsQOAopwc5paXHxNH0dPZ2vsmQOrl7RPhzx7B0VkVzbVbKBh3pOx9XXs7f6/aTWVL67HnbQniiwuC1zghqeojUy5mwYWTCXjS05DY5QiuOJyl+CyLmydN5uZJzo18sKuLJzZu4K97dkPARAJObIHGFLpjcbGw2dfezu/WH6mifHHhKCZG6xl+JDcKy5O+vIoTiLszW5o2sSri55364zwXJhheZ2Zg+IzDIc29XFU6llsvnsaYHHcPIdO44jBEGBkI8PnLLufzl10OwLaDjfx2/To2NTRAtoVkO/9qDTv7FL37FRsPNLCmqxnDmMGlw/YzMzc9FaJPRsQSQlkGzVTxTn2xs1TwHlkq9KZI93L+sOEsmjaDaaPTV9TGpW+44jBEuXBkAd+/7gYAbFWWVVXy2/XraKIb8R67X0GDQhRWdZ1HMC/Cf72xLCM23VpqYlkx/L4IeVYPFh7Ee+y+QcDysGj6DG4oPx/LDWc+o7jicA5giHBtWTnXljmZil2RCH/cttVxh2aZWAXD0dg+DIHtvjKsoswsK17WmXxx4p8QheU7pmP4nJv/pomT+PDkKeT6Mriccek3ku4MuVSYOXOmrl59QskHlwxT397OS6tepinUwrJWi0wGzuZLLROzA9x25ccYm59/+gNc0o6IrFHV0zZUcWcOLpyXk8Mn5s4H4Ctn2BaXwYO7qHNxcUmIKw4uLi4JccXBxcUlIa44uLi4JMQVBxcXl4S44uDi4pIQVxxcXFwSMqiCoESkEag+03ZkgJFA/1tcnR2453b2MVZVT1t/f1CJw1BFRFb3JSLtbMQ9t6GLu6xwcXFJiCsOLi4uCXHFYWA4eX+2sx/33IYo7p6Di4tLQtyZg4uLS0JcccgQIvJhEdksIraIzDzuubtFZJeIbBeRd58pG1NFRN4TP4ddIvL1M21PKojIr0WkQUQ2HfXYcBH5q4jsjH8fdiZtHGhcccgcm4CbgdeOflBEJgMLgSnAe4CfiYg58OalRtzm/wTeC0wGbomf29nKIzj/j6P5OvCyqk4AXo7/fs7gikOGUNWtqro9wVM3AU+qakhVK4FdQMXAWpcWKoBdqrpHVcPAkzjndlaiqq8Bzcc9fBPwaPznR4H5A2rUGcYVh4GnCNh71O+18cfONobKeZyKQlWtB4h/H3WG7RlQ3DJxKSAifwNGJ3jqG6r6/MkOS/DY2egyGirn4XISXHFIAVW9PonDaoGSo34vBurSY9GAMlTO41QcEJHzVLVeRM4DGs60QQOJu6wYeJYAC0XEJyJlwATg7TNsUzKsAiaISJmIeHE2WZecYZvSzRJgUfznRcDJZoNDElccMoSILBCRWmA2sFREXgJQ1c3A08AW4EXgs6oaO3OWJoeqRoHPAS8BW4Gn4+d2ViIiTwArgIkiUisinwbuB24QkZ3ADfHfzxncCEkXF5eEuDMHFxeXhLji4OLikhBXHFxcXBLiioOLi0tCXHFwcXFJiCsOLi4uCXHFwcXFJSGuOLi4uCTkfwEf6LBZc/CsAQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, mesh = sheet_view(\n",
    "    sheet, mode=\"2D\", \n",
    "    edge={\"color\": np.abs(phi), \"width\": 3},\n",
    "    face={\"visible\":True, \"color\": sheet.face_df[\"theta\"], \"alpha\": 0.3})\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}