{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# EE16A Discussion 2B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing Matrices as Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This exercise is going to help you visualize matrices as operations. For example, when we multiply a vector by a rotation matrix, we will see it \"rotate\" in the true sense here. Similarly, when we multiply a matrix by a scalar matrix, we will see it \"scale\". The way we will see this is by applying the operation to all the vertices of a polygon and seeing how the polygon changes.\n",
    "Let's first do the necessary imports and define some useful functions to do this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\"\"\"Function that takes the vertices of a polygon and applies a matrix \"transformation\" to each of them, effectively\n",
    "\"transforming\" the polygon.\"\"\"\n",
    "def transform_the_polygon(polygon, T):\n",
    "    \n",
    "    transformed_polygon = []\n",
    "    for point in polygon:\n",
    "        transformed_point = np.dot(T, point)\n",
    "        transformed_polygon.append(transformed_point)\n",
    "    return transformed_polygon\n",
    "\n",
    "\"\"\"Function that plots a polygon in the x-y plane, given its vertices as x-y coordinates. The plot is defined in terms\n",
    "of line segments connecting all adjacent vertices of the polygon.\"\"\"\n",
    "def plot_the_polygon(polygon):\n",
    "    fig = plt.figure(figsize=(5,5))\n",
    "    ax = fig.add_subplot(111, xlim = [-4, 4], ylim = [-4, 4])\n",
    "    for i in range(len(polygon) - 1):\n",
    "        ax.plot([polygon[i][0], polygon[i+1][0]],\n",
    "                [polygon[i][1], polygon[i+1][1]], linewidth=4)\n",
    "    ax.plot([polygon[i+1][0], polygon[0][0]], [polygon[i+1][1], polygon[0][1]], linewidth=4)\n",
    "    ax.grid(True)\n",
    "    ax.axhline(y=0, color='k', linestyle = '--', linewidth = 2)\n",
    "    ax.axvline(x=0, color='k', linestyle = '--', linewidth = 2)\n",
    "    #plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define our starting polygon, a square whose side is of length 1. Let's see what the square looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "unit_square = [np.array([0,0]), np.array([1,0]), np.array([1,1]), np.array([0,1])]\n",
    "plot_the_polygon(unit_square)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part 1: Rotation Matrices as Rotations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Code for the following sections:\n",
    "\n",
    "(a) Given T1 (15 degree rotation) and T2 (30 degree rotation), describe how to rotate the unit square by 45 degrees.  How about 60 degrees?\n",
    "\n",
    "(b) Try to rotate the unit squre by 60 degrees using only one matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# R is a \"rotation matrix.\"\n",
    "# Example:\n",
    "# angle = np.pi/2 # 90 degrees in radians\n",
    "# R = np.array([[np.cos(angle), -np.sin(angle)],\n",
    "#              [np.sin(angle), np.cos(angle)]])\n",
    "\n",
    "angle1 = np.pi/12 # 15 degrees in radians\n",
    "T1 = np.array([[np.cos(angle1), -np.sin(angle1)],\n",
    "              [np.sin(angle1), np.cos(angle1)]])\n",
    "\n",
    "angle2 = np.pi/6 # 30 degrees in radians\n",
    "T2 = np.array([[np.cos(angle2), -np.sin(angle2)],\n",
    "              [np.sin(angle2), np.cos(angle2)]])\n",
    "\n",
    "# fill in as appropriate\n",
    "T3 = np.matmul(T1,T2)\n",
    "\n",
    "rotated_square = transform_the_polygon(unit_square, T3)\n",
    "plot_the_polygon(rotated_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And finally, we can \"reflect\" the square about the y-axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Re_y = np.array([[-1, 0], [0,1]])\n",
    "reflected_square = transform_the_polygon(unit_square, Re_y)\n",
    "plot_the_polygon(reflected_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part 2: Commutativity of Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The next natural question to ask is the following: Does the *order* in which you apply these operations matter?\n",
    "\n",
    "a) Let's see what happens to the unit square when we rotate the matrix by 60 degrees, and then reflect it along the y-axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# As the name indicates, R_60 rotates the matrix by 60 degrees.\n",
    "rotation_angle = np.pi/3 # 60 degrees in radians\n",
    "R_60 = np.array([[np.cos(rotation_angle), -np.sin(rotation_angle)],\n",
    "                 [np.sin(rotation_angle), np.cos(rotation_angle)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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t/tymaOxOEi+e4hKNRqn66Avu//1SXnmuir17Oh6aD83YydhJJ/R6bZJ8qXEnuwROTp9M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\nIoH2/wFQ/NDqvtKvdgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "reflected_after_rotated_square = transform_the_polygon(unit_square, R_60)\n",
    "reflected_after_rotated_square = transform_the_polygon(reflected_after_rotated_square, Re_y)\n",
    "plot_the_polygon(reflected_after_rotated_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b) Now, let's reflect *before* rotating."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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2koZ+j3brazv4OkTGWrsV2Aic0GPyiUD3dMkRa++6h8TO3VdvlB4+iFGXXBxgRckeem4z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\nYII=\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rotated_after_reflected_square = transform_the_polygon(unit_square, Re_y)\n",
    "rotated_after_reflected_square = transform_the_polygon(rotated_after_reflected_square, R_60)\n",
    "plot_the_polygon(rotated_after_reflected_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also scale the square: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# S is a \"scalar matrix.\"\n",
    "S = np.array([[2, 0],\n",
    "              [0, 2]])\n",
    "scaled_square = transform_the_polygon(unit_square, S)\n",
    "plot_the_polygon(scaled_square)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernel_info": {
   "name": "python3"
  },
  "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": "3.6.5"
  },
  "nteract": {
   "version": "0.12.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}