Transformation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transformation matrices\n",
    "\n",
    "### A tutorial on different transformation matrix from a computational scientist point of view"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#importing necessary modules\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Define x and y coordinates of entire reference domain\n",
    "x_cor = np.linspace(1,10,20)\n",
    "y_cor = np.linspace(1,10,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define H\n",
    " Define any shape with which you would like to play...:) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = []\n",
    "#Finding boundaries of the reference domain\n",
    "min_x_cor = np.min(x_cor)\n",
    "max_x_cor = np.max(x_cor)\n",
    "min_y_cor = np.min(y_cor)\n",
    "max_y_cor = np.max(y_cor)\n",
    "\n",
    "#Iterative loop with conditional selection of required points in the domain\n",
    "for i in range(len(x_cor)):\n",
    "    for j in range(len(y_cor)):\n",
    "        if (x_cor[i] < min_x_cor+2 or x_cor[i] > max_x_cor-2) or (y_cor[j] > min_y_cor+3 and y_cor[j] < max_y_cor-3):\n",
    "            x.append([x_cor[i],y_cor[j]])    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x238be277548>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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VtXoPOLnPf/W+Bav0ANA5VukBYMYwwAGgKAY4ABTFAAeAohjgAFDUhr4LJSJWJB3esC94YeyQ9Bv3RXSE5+MMnouz8XycMelz8YeZuXP9gxs6wGdBRCy/3tt5NiuejzN4Ls7G83HGhXouuIUCAEUxwAGgKAb46JbcF9AZno8zeC7OxvNxxgV5LrgHDgBF8TdwACiKAQ4ARTHAG0XEOyLihxFxKCKeiYg73dfkFhFbIuLJiPiO+1rcImJ7ROyLiB8Pzsj73NfkEhGfGPw78nREPBwRb3Rf00aKiC9HxImIeHrNY2+NiEcj4qeDXy+ZxtdigLd7VdInM/NPJF0j6e8j4l3ma3K7U9Ih90V04guSvpeZV0p6jzbp8xIRl0v6mKSFzLxK0hZJt3qvasN9RdIH1j12t6THMvOPJT02+PPEGOCNMvN4Zj4x+P2LWv0XdNP+ZIWI2CXpg5IecF+LW0RcLOkvJX1JkjLz95l50ntVVlslzUXEVkkXSXrefD0bKjP/Q9Jv1z18k6SvDn7/VUk3T+NrMcDHEBHzkq6W9Lj3Sqw+L+lTkv7HfSEd+CNJK5L+ZXBL6YGIeJP7ohwy85ikz0o6Ium4pBcy8/veq+rC2zPzuLT6l0FJb5vGJ2WAjygi3izpG5I+npm/c1+PQ0TcKOlEZh5wX0sntkp6r6QvZubVkl7SlP4TuZrBvd2bJL1T0mWS3hQRt3mvanYxwEcQEdu0Orwfysz97usxulbShyLiF5K+Jun9EfGg95Ksjko6mpmn/4tsn1YH+mb0V5J+npkrmfmKpP2S/sJ8TT34dURcKkmDX09M45MywBtFRGj1HuehzPyc+3qcMnNvZu7KzHmtvkD1g8zctH/LysxfSfplRJz+cePXS3rWeElORyRdExEXDf6duV6b9AXddb4t6fbB72+X9K1pfNKufyp9Z66V9BFJByPiqcFjn87M7xqvCf34B0kPRcQbJP1M0kfN12ORmY9HxD5JT2j1nVtPapOt1EfEw5Kuk7QjIo5KukfSfZK+HhF3aPX/5G6ZytdilR4AauIWCgAUxQAHgKIY4ABQFAMcAIpigANAUQxwACiKAQ4ARf0vAWIz03BjLHgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plotting the reference configuration of object\n",
    "x = np.array(x)\n",
    "plt.scatter(x[:,0],x[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear transformation\n",
    "\n",
    "Define a matrix that can move our H by some distance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5, 0]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Let's define transformation matrix\n",
    "#Our matrix should move each point in x by 5 units\n",
    "move = np.array([5,0])\n",
    "move = move.reshape(1,-1) #reshape matrix to desired shape\n",
    "move"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "shape of moving matrix: (1, 2)\n",
      "shape of reference configuration matrix: (260, 2)\n"
     ]
    }
   ],
   "source": [
    "#Now let's check the dimensions of our matrix\n",
    "print(\"shape of moving matrix: {}\".format(move.shape))\n",
    "print(\"shape of reference configuration matrix: {}\".format(x.shape))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Finally let's move the matrix\n",
    "#we will explore the power of numpy's broadcasting capabilities\n",
    "x_move  = x + move"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x238bf309108>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "#Now let's plot the shapes\n",
    "plt.scatter(x[:,0],x[:,1], color = \"k\")\n",
    "plt.scatter(x_move[:,0],x_move[:,1], color = \"b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stretching matrix\n",
    "\n",
    "Matrix that can stretch the given reference configuration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.5, 0. ],\n",
       "       [0. , 1. ]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#lets define a stretching matrix\n",
    "#Our matrix should stretch H in x-direction by about 50%\n",
    "#Think that you are applying longitudinal strain\n",
    "stretch = np.array([[1.5,0],[0,1]])\n",
    "stretch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "shape of stretch matrix: (2, 2)\n",
      "shape of reference configuration (H) matrix: (260, 2)\n"
     ]
    }
   ],
   "source": [
    "print(\"shape of stretch matrix: {}\".format(stretch.shape))\n",
    "print(\"shape of reference configuration (H) matrix: {}\".format(x.shape))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Now we have all the ingredients, let's stretch\n",
    "#x_stretch = x*stretch (each point should be multiplied by stretch matrix)\n",
    "x_stretch = np.dot(x,stretch) #Matrix multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "shape of reference configuration (H) matrix: (260, 2)\n",
      "shape of stretched configuration (H) matrix: (260, 2)\n"
     ]
    }
   ],
   "source": [
    "print(\"shape of reference configuration (H) matrix: {}\".format(x.shape))\n",
    "print(\"shape of stretched configuration (H) matrix: {}\".format(x_stretch.shape))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x238bf37c948>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "#plot the reference configuration and stretched configuration\n",
    "plt.scatter(x[:,0],x[:,1],color=\"k\")\n",
    "plt.scatter(x_stretch[:,0],x_stretch[:,1],color=\"b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotation matrix\n",
    "\n",
    "Matrix that can rotate the reference configuration by some angle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.8660254,  0.5      ],\n",
       "       [-0.5      ,  0.8660254]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Define matrix that can rotate the reference configuration\n",
    "theta = 30*np.pi/180  #Angle to rotate\n",
    "rotate = np.array([[np.cos(theta),np.sin(theta)],[-np.sin(theta),np.cos(theta)]])\n",
    "rotate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Now we have all the ingredients, let's rotate the matrix\n",
    "#x_rotate = x*stretch (each point should be multiplied by stretch matrix)\n",
    "x_rotate = np.dot(x,rotate) #Matrix multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x238bf3e9088>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "#plotting the reference configuration and rotated matrix\n",
    "plt.scatter(x[:,0],x[:,1],color=\"k\")\n",
    "plt.scatter(x_rotate[:,0],x_rotate[:,1],color=\"b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shear matrix\n",
    "\n",
    "Matrix that can shear reference configuration by some angle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 0. ],\n",
       "       [0.5, 1. ]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Let's define the shear matrix\n",
    "shear = np.array([[1,0],[0.5,1]])\n",
    "shear"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Now we have all the ingredients, let's shear the matrix\n",
    "#x_shear = x*shear (each point should be multiplied by stretch matrix)\n",
    "x_shear = np.dot(x,shear) #Matrix multiplication"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x238bf459808>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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9uJ/pAADaWvUd+AclvVOz801JZnbUzE6b2ekLFy50foIoScy2fQ1LptxIxDUr2Se17bodU09M9CRVGG9zSbpZ0p/Nf/1SSZ/a7WvGegql7Wc6l/wJP6cBmpX8jPi263ZMnweO7tT3KRRJvy/pvKQHJX1b0g8lnVz2NeNNYp5d+MaaXbnTazkSdlOUZ030u27H1BMT3TVt4L0kMc3spZLe4T2fQomTxGzX17Bkyo1EXLOyfVLbrdsx9cREdyQxM92zS1/Dkik3EnHNyvZJbTefMfXERI9Sb8tzXdTAqYHXiBr48OsR3YgkZr57dulrWDLlRiKuWck+qW3X7Zh6YqKbpg286k8jBAAErYEDAJqxgQNAUGzgABBU9Rt4jVH6VRrTpp+H2HxptUXpx9TUGBmlfrKZ6xrDMcJVPpSfI4N1qu0Y4ZgaOqAfiniMsMYo/aqNaYnN16e2KP2YmhqjH00beNXHCGuM0pdqOEtsPp/aovRjamqMfoQ8RlhjlH7VxrTE5utTW5R+TE2NkVnqbXmuixo4NfAaUQPvPpY1NixFrIG71xmlX4wLd2lMm34eYvOl1RalT/3vF7WpMVbXtIFXXQMHAAStgQMAmrGBA0BQ1W/gJZOY7ZvIxmhqjGYlk5ht106EpsYYWKownuuKdAql/Qfox2jogGYlT6G0XTsRGjogH0U8hVIyidm+ieymR2hqjGZlk5jt1k6EpsbIp2kDr/oUSskkZvsmsjGaGqNZ2SRmu7UToakx8gl5CqVkErN9E9kYTY3RrGwSs93aidDUGAWk3pbnuqiBUwOvETVw1lntFLEG7l42idm+iWyMpsZoVjKJ2XbtRGhqjDyaNvCqa+AAgKA1cABAMzZwAAiq+g18qCRm2x6Es6+vvycmuinV67LL2qmtJyYqkCqM57pqPYWS+lzl1OcvR/k8cHRT8nO+266d2j4PHMNSxFMowyUxzy58U12+zu4YG6EnJroZKom56tqpqScmhtW0gVd9CmW4JGbbHoRdxpbriYluovS6rKknJoYV8hTKcEnMdBou9XiEnpjoprZelxF6YqISqbfluS5q4NTAa0QNfG9jMRxFrIG7D5fETCXNIvfERDelel12WTu19cTEcJo28Kpr4ACAoDVwAECzPW/gZvZsM/ucmZ0xsy+b2S19TgwAsNwq78AvSnq7uz9P0o2S3mxmz+9nWk/IkcRs21swQk9M9KNUr8soPTFRqVRhfC+XpE9KetmyMTWcQmn7ucoRPg8c/Sj1Od9RPg8c5SnnKRRJW5odSn1a4s+OSjot6fTGxkanSedIYrbtLRihJyb6kSeJufua6LJ2SvbERHlNG/jKp1DM7KmS/knSCXf/+LKxNSQx2/YWjNATE/0o1esySk9MlJflFIqZHZT0MUmndtu89yJHErNtb8EIPTHRj1K9LqP0xETFUm/L21ySTNJfSfpg26+hBk4NvEbUwFlntVPfNXBJvzhfxPdL+sL8esWyr6klidm2t2CEnpjoR6lel1F6YqKspg2cJCYAVI4kJgCMDBs4AARV/Qbedw/CLr0Fa+uJiXxK9bqssScmAkkVxnNdQ59CWeVzlWv7PHDkU/Jzvmv7PHDUSRE/D3yoHoQRemIin6HWWYSemKhT0wZe9SmU+noQrn5PUpf1qW+dscZwpZCnUIbqQRihJybyqa3XZcmemIil6g38xIkTWl9fv+Kx9fV1nThxotW4o0cf1MLDWlu7qIMH33vFYwcPvldraxd3Hdd0T+kxSb+z67j1dWlh6jpxQq3GIZ8c66ztmmg7ru26ZY1NTKqukusqkcRcpbdgbT0xkU+pXpc19sREfRSxBg4ACFoDBwA0YwMHgKCq38D77kEoxeiJiWG1XWd9r4nSPTERXKownuvKmcTs+3OVS34eOIbVdp31vSZKfx444tD4k5hnW6XNIvTExLDarrO+10TpnpiIo2kDr/oUSt89CGe/r78nJobVdp31vSZK98REHCFPofTdg1CK0RMTw2q7zvpeE6V7YiK+qjfwLgm5WUrtsSseX1u7uCNtdvTogzvGSY/NH+8+rstzk4irU9t11veayLFuWWMTk6qr5LpyJzH77i1YsicmhtV2nfW9Jkr3xEQMilgDBwAErYEDAJqxgQNAUGzgABAUGzgABMUGDgBBsYEDQFBs4AAQFBs4AAQ1aJDHzC5IOjfYE7ZzSNJ3S0+iJeaaR6S5SrHmy1z7senuhxcfHHQDr5GZnU4lnGrEXPOINFcp1nyZa16UUAAgKDZwAAiKDVy6rfQEOmCueUSaqxRrvsw1o8nXwAEgKt6BA0BQbOAAENQkN3Aze7aZfc7MzpjZl83sltJz2o2Z7TezfzOzT5Wey27M7Cozu9PMvjr/O35x6Tk1MbO3zdfAA2Z2h5k9qfScLjOzj5jZI2b2wLbHnm5md5nZ1+f/vbrkHC9rmOsH5mvgfjP7WzO7quQct0vNd9ufvcPM3MwOlZhbF5PcwCVdlPR2d3+epBslvdnMnl94Tru5RdKZ0pNo6U8k/aO7P1fSC1TpvM3sWZLeKumIu18nab+kXy07qyvcLunlC4/dKukz7v6zkj4z/30NbtfOud4l6Tp3/zlJ/y7pXUNPaonbtXO+MrNnS3qZmrpNV2aSG7i7f8vd75v/+geabTDPKjurZmZ2raRXSvpw6bnsxsyeJuklkv5Cktz9f939+2VntdQBSU82swOS1iWl278X4O7/LOl7Cw+/WtJH57/+qKRfGXRSDVJzdfdPu/vF+W//VdK1g0+sQcPfrST9saR3SgpxumOSG/h2ZrYl6UWSPl92Jkt9ULNFdan0RFr4aUkXJP3lvOTzYTN7SulJpbj7NyT9oWbvtr4l6T/d/dNlZ7WrH3f3b0mzNyKSnlF4Pm39hqR/KD2JZczsVZK+4e5fLD2Xtia9gZvZUyV9TNJvuft/lZ5PipndLOkRd7+39FxaOiDpekkfcvcXSXpM9fwz/wrz+vGrJf2UpJ+U9BQze13ZWY2Pmb1bs7LlqdJzaWJm65LeLel3S8+li8lu4GZ2ULPN+5S7f7z0fJb4BUmvMrMHJf21pF8ys5Nlp7TUeUnn3f3yv2ju1GxDr9EvSzrr7hfc/XFJH5f084XntJvvmNkzJWn+30cKz2cpM3uDpJsl/ZrXHTr5Gc3+j/yL8++1ayXdZ2Y/UXRWu5jkBm5mplmN9oy7/1Hp+Szj7u9y92vdfUuzH7B91t2rfZfo7t+W9LCZPWf+0E2SvlJwSss8JOlGM1ufr4mbVOkPXLf5O0lvmP/6DZI+WXAuS5nZyyX9tqRXufsPS89nGXf/krs/w9235t9r5yVdP1/P1ZrkBq7Zu9rXa/Zu9gvz6xWlJzUib5F0yszul/RCSb9XeD5J838l3CnpPklf0uz7oZo4tZndIelfJD3HzM6b2ZskvU/Sy8zs65qdlnhfyTle1jDXP5X0Y5Lumn+P/XnRSW7TMN9wiNIDQFBTfQcOAOGxgQNAUGzgABAUGzgABMUGDgBBsYEDQFBs4AAQ1P8BgtTMxkS+JKgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot the sheared configuration and reference configuration\n",
    "plt.scatter(x[:,0],x[:,1],color=\"k\")\n",
    "plt.scatter(x_shear[:,0],x_shear[:,1],color=\"b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Will keep improving this notebook to include other transformation matrices <br>\n",
    "## Thanks you..!"
   ]
  },
  {
   "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": "3.7.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}