{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# EECS16A Discussion 1B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving Systems of Equations: Visualization\n",
    "### Initialization\n",
    "First, run this command in your terminal to install some dependencies:\n",
    "\n",
    "    pip install mpld3 plotly\n",
    "\n",
    "Then run the code in the next block to set up functions that the other blocks\n",
    "depend on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "        <script type=\"text/javascript\">\n",
       "        window.PlotlyConfig = {MathJaxConfig: 'local'};\n",
       "        if (window.MathJax) {MathJax.Hub.Config({SVG: {font: \"STIX-Web\"}});}\n",
       "        if (typeof require !== 'undefined') {\n",
       "        require.undef(\"plotly\");\n",
       "        requirejs.config({\n",
       "            paths: {\n",
       "                'plotly': ['https://cdn.plot.ly/plotly-latest.min']\n",
       "            }\n",
       "        });\n",
       "        require(['plotly'], function(Plotly) {\n",
       "            window._Plotly = Plotly;\n",
       "        });\n",
       "        }\n",
       "        </script>\n",
       "        "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.7/site-packages/plotly/graph_objs/_deprecations.py:550: DeprecationWarning:\n",
      "\n",
      "plotly.graph_objs.XAxis is deprecated.\n",
      "Please replace it with one of the following more specific types\n",
      "  - plotly.graph_objs.layout.XAxis\n",
      "  - plotly.graph_objs.layout.scene.XAxis\n",
      "\n",
      "\n",
      "/usr/local/lib/python3.7/site-packages/plotly/graph_objs/_deprecations.py:578: DeprecationWarning:\n",
      "\n",
      "plotly.graph_objs.YAxis is deprecated.\n",
      "Please replace it with one of the following more specific types\n",
      "  - plotly.graph_objs.layout.YAxis\n",
      "  - plotly.graph_objs.layout.scene.YAxis\n",
      "\n",
      "\n",
      "/usr/local/lib/python3.7/site-packages/plotly/graph_objs/_deprecations.py:603: DeprecationWarning:\n",
      "\n",
      "plotly.graph_objs.ZAxis is deprecated.\n",
      "Please replace it with one of the following more specific types\n",
      "  - plotly.graph_objs.layout.scene.ZAxis\n",
      "\n",
      "\n",
      "/usr/local/lib/python3.7/site-packages/plotly/graph_objs/_deprecations.py:494: DeprecationWarning:\n",
      "\n",
      "plotly.graph_objs.Scene is deprecated.\n",
      "Please replace it with one of the following more specific types\n",
      "  - plotly.graph_objs.layout.Scene\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import mpld3\n",
    "import numpy.matlib\n",
    "import matplotlib.pyplot as plt\n",
    "import chart_studio.plotly as py\n",
    "import plotly.offline as offline\n",
    "from plotly.graph_objs import *\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from numpy import pi, cos, exp\n",
    "\n",
    "%matplotlib inline\n",
    "mpld3.enable_notebook()\n",
    "# offline.init_notebook_mode()\n",
    "offline.init_notebook_mode(connected=True)\n",
    "\n",
    "# Define the function to be plotted\n",
    "def fxy(x, y):    \n",
    "    A = 1  # choose a maximum amplitude \n",
    "    return A*(cos(pi*x*y))**2 * exp(-(x**2+y**2)/2.)\n",
    "\n",
    "def line_xy(a,b,c):\n",
    "    '''For the parameters a,b,c this function returns an array of (x,y) pairs\n",
    "    that are on the line ax+by = c'''\n",
    "    b = b+.00001\n",
    "    x_range = [-100, 100]\n",
    "    x = np.linspace(x_range[0],x_range[1],100)\n",
    "    y = (c - a*x) / b    \n",
    "    return x, y\n",
    "\n",
    "def plane_xyz(a,b,c,d):\n",
    "    '''For the parameters a,b,c,d this function returns an array of (x,y,z) pairs\n",
    "    that are on the line ax+by+cz = d'''\n",
    "    \n",
    "    c = c+.00001\n",
    "    x_range = [-10, 10]\n",
    "    x = np.linspace(x_range[0],x_range[1],100)\n",
    "    y = np.linspace(x_range[0],x_range[1],100)\n",
    "    yt = y[:,np.newaxis]\n",
    "    \n",
    "    x_mat = np.matlib.repmat(x,len(y),1)\n",
    "    y_mat = np.matlib.repmat(yt,1,len(x))\n",
    "    z_mat = (d - a*x_mat - b*y_mat) / c\n",
    "#   for k in range(0,len(zpoints)): \n",
    "#           x_short,y_short=line_xy(a,b,c*zpoints[k]+d)\n",
    "#           z_short=np.ones(len(y_short))*zpoints[k]\n",
    "#           x=np.append(x,x_short)\n",
    "#           y=np.append(y,y_short)\n",
    "#           z=np.append(z,z_short)\n",
    "    return x, y, z_mat\n",
    "\n",
    "axis = dict(\n",
    "    showbackground=True,                  # (!) show axis background\n",
    "    backgroundcolor=\"rgb(204, 204, 204)\", # set background color to grey\n",
    "    gridcolor=\"rgb(255, 255, 255)\",       # set grid line color\n",
    "    zerolinecolor=\"rgb(255, 255, 255)\",   # set zero grid line color\n",
    "    range=[-10, 10]\n",
    ")\n",
    "\n",
    "# Make a layout object\n",
    "layout = Layout(\n",
    "    title='Planes in 3D',   # set plot title\n",
    "    scene=Scene(            # (!) axes are part of a 'scene' in 3d plots\n",
    "        xaxis=XAxis(axis),  # set x-axis style\n",
    "        yaxis=YAxis(axis),  # set y-axis style\n",
    "        zaxis=ZAxis(axis),  # set z-axis style\n",
    "    ),\n",
    "    showlegend=False,\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graphing System of Two Equations\n",
    "\n",
    "We now take a graphical interpretation of a system of two equations in two variables. Each equation can be rewritten as $ax+by=c$ and thus describes a line in $\\mathbb{R}^2$. Therefore, the solution to the system of equations must lie on the lines described by the system of equations.\n",
    "\n",
    "In the next block, there is code to graph two lines. Play around with the parameters $a1,b1,\\ldots,c2$ to create different lines, and note that the point of intersection for the two lines is a unique solution satisfying both equations. Use the graph to explain the results you obtained in Problem 1 part (a) and (c)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10ccba610>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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       "\n",
       "\n",
       "<style>\n",
       "\n",
       "</style>\n",
       "\n",
       "<div id=\"fig_el1660244479739044440975274\"></div>\n",
       "<script>\n",
       "function mpld3_load_lib(url, callback){\n",
       "  var s = document.createElement('script');\n",
       "  s.src = url;\n",
       "  s.async = true;\n",
       "  s.onreadystatechange = s.onload = callback;\n",
       "  s.onerror = function(){console.warn(\"failed to load library \" + url);};\n",
       "  document.getElementsByTagName(\"head\")[0].appendChild(s);\n",
       "}\n",
       "\n",
       "if(typeof(mpld3) !== \"undefined\" && mpld3._mpld3IsLoaded){\n",
       "   // already loaded: just create the figure\n",
       "   !function(mpld3){\n",
       "       \n",
       "       mpld3.draw_figure(\"fig_el1660244479739044440975274\", {\"width\": 432.0, \"height\": 288.0, \"axes\": [{\"bbox\": [0.125, 0.125, 0.775, 0.755], \"xlim\": [0.0, 5.0], \"ylim\": [0.0, 5.0], \"xdomain\": [0.0, 5.0], \"ydomain\": [0.0, 5.0], \"xscale\": \"linear\", \"yscale\": \"linear\", \"axes\": [{\"position\": \"bottom\", \"nticks\": 6, \"tickvalues\": null, \"tickformat\": null, \"scale\": \"linear\", \"fontsize\": 10.0, \"grid\": {\"gridOn\": true, \"color\": \"#B0B0B0\", \"dasharray\": \"none\", \"alpha\": 1.0}, \"visible\": true}, {\"position\": \"left\", \"nticks\": 6, \"tickvalues\": null, \"tickformat\": null, \"scale\": \"linear\", \"fontsize\": 10.0, \"grid\": {\"gridOn\": true, \"color\": \"#B0B0B0\", \"dasharray\": \"none\", \"alpha\": 1.0}, \"visible\": true}], \"axesbg\": \"#FFFFFF\", \"axesbgalpha\": null, \"zoomable\": true, \"id\": \"el166024447973584\", \"lines\": [{\"data\": \"data01\", \"xindex\": 0, \"yindex\": 1, \"coordinates\": \"data\", \"id\": \"el166024509548432\", \"color\": \"#0000FF\", \"linewidth\": 1.5, \"dasharray\": \"9.600000000000001,2.4000000000000004,1.5,2.4000000000000004\", \"alpha\": 1, \"zorder\": 2, \"drawstyle\": \"default\"}, {\"data\": \"data01\", \"xindex\": 0, \"yindex\": 2, \"coordinates\": \"data\", \"id\": \"el166024509219856\", \"color\": \"#FF0000\", \"linewidth\": 1.5, \"dasharray\": \"none\", \"alpha\": 1, \"zorder\": 2, \"drawstyle\": \"default\"}, {\"data\": \"data02\", \"xindex\": 0, \"yindex\": 1, \"coordinates\": \"axes\", \"id\": \"el166024509642704\", \"color\": \"#0000FF\", \"linewidth\": 1.5, \"dasharray\": \"9.600000000000001,2.4000000000000004,1.5,2.4000000000000004\", \"alpha\": 1, \"zorder\": 1000002.0, \"drawstyle\": \"default\"}, {\"data\": \"data02\", \"xindex\": 0, \"yindex\": 2, \"coordinates\": \"axes\", \"id\": \"el166024509710096\", \"color\": \"#FF0000\", \"linewidth\": 1.5, \"dasharray\": \"none\", \"alpha\": 1, \"zorder\": 1000002.0, \"drawstyle\": \"default\"}], 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       "   }(mpld3);\n",
       "}else if(typeof define === \"function\" && define.amd){\n",
       "   // require.js is available: use it to load d3/mpld3\n",
       "   require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
       "   require([\"d3\"], function(d3){\n",
       "      window.d3 = d3;\n",
       "      mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3.js\", function(){\n",
       "         \n",
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       "      });\n",
       "    });\n",
       "}else{\n",
       "    // require.js not available: dynamically load d3 & mpld3\n",
       "    mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
       "         mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3.js\", function(){\n",
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       "            })\n",
       "         });\n",
       "}\n",
       "</script>"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "# use for parts i-iii\n",
    "# Parameters for the first line\n",
    "a1 = 2\n",
    "b1 = 1\n",
    "c1 = 6\n",
    "\n",
    "# Parameters for the second line\n",
    "a2 = 3\n",
    "b2 = -2\n",
    "c2 = 2\n",
    "\n",
    "x1, y1 = line_xy(a1,b1,c1)\n",
    "x2, y2 = line_xy(a2,b2,c2)\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111,xlim=[0,5],ylim=[0,5])\n",
    "ax.plot(x1,y1,'-.b')\n",
    "ax.plot(x2,y2,'r')\n",
    "ax.grid(True)\n",
    "ax.legend(['Line 1','Line 2'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graphing System of Three Equations\n",
    "\n",
    "For a system of three equations, each equation can be rewritten as $ax+by+cz=d$. This is the equation for a plane in $\\mathbb{R}^3$. As you might have already guessed, the solution for the system of equations must lie on all of the planes described by the system.\n",
    "\n",
    "In the next block, there is code to graph planes. Play around with the parameters $a1,b1,\\ldots,d3$ to create different planes. Use the graph to explain the results you obtained in Problem 1 part (b).\n",
    "\n",
    "**Rotate the figures to make sure you see all the graphs. It will take a few seconds for the figure to load.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(100,)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.7/site-packages/plotly/graph_objs/_deprecations.py:40: DeprecationWarning:\n",
      "\n",
      "plotly.graph_objs.Data is deprecated.\n",
      "Please replace it with a list or tuple of instances of the following types\n",
      "  - plotly.graph_objs.Scatter\n",
      "  - plotly.graph_objs.Bar\n",
      "  - plotly.graph_objs.Area\n",
      "  - plotly.graph_objs.Histogram\n",
      "  - etc.\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
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       "config": {
        "linkText": "Export to plot.ly",
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       "                        {\"responsive\": true}\n",
       "                    ).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('1add8fe8-0c8b-4197-a5a6-c1a79eb75a97');\n",
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       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
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       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
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       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
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       "\n",
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       "            </script>\n",
       "        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Use for parts iv-vi\n",
    "\n",
    "# Parameters for the first plane\n",
    "a1 = 1\n",
    "b1 = 1\n",
    "c1 = 1\n",
    "d1 = 2\n",
    "\n",
    "# Parameters for the second plane\n",
    "a2 = 1\n",
    "b2 = -1\n",
    "c2 = 0\n",
    "d2 = 1\n",
    "\n",
    "# Parameters for the third plane\n",
    "a3 = 0\n",
    "b3 = 2\n",
    "c3 = 1\n",
    "d3 = 1\n",
    "\n",
    "# Plotting \n",
    "x1, y1, z1 = plane_xyz(a1,b1,c1,d1)\n",
    "x2, y2, z2 = plane_xyz(a2,b2,c2,d2)\n",
    "x3, y3, z3 = plane_xyz(a3,b3,c3,d3)\n",
    "\n",
    "print(x1.shape)\n",
    "blue=[[0, 'rgb(0,0,255)'], [1, 'rgb(0,0,255)']]\n",
    "green=[[0, 'rgb(0,255,0)'], [1, 'rgb(0,255,0)']]\n",
    "red=[[0, 'rgb(255,0,0)'], [1, 'rgb(255,0,0)']]\n",
    "\n",
    "trace1 = Surface(z=z1, x=x1,y=y1, colorscale=blue,name='First Plane')\n",
    "trace2 = Surface(z=z2, x=x2,y=y2, colorscale=green, name='Second Plane')\n",
    "trace3 = Surface(z=z3, x=x3,y=y3, colorscale=red, name='Third Plane')\n",
    "data = Data([trace1,trace2, trace3])\n",
    "fig = Figure(data=data,layout=layout)\n",
    "offline.iplot(fig, filename='planes')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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