Lecture06.ipynb

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   "source": [
    "# Lecture 6\n",
    "\n",
    "Using partial differentiation, we showed in the lecture that for the implicit representation of a line\n",
    "\n",
    "$$\n",
    "f(x, y) = c,\n",
    "$$\n",
    "\n",
    "where $f$ is and function and $c$ is constant, the gradient $\\nabla f$  on the line is perpendicular to the line. We demonstrate this here for an ellipse.\n",
    "\n",
    "## Equation of an ellipse\n",
    "\n",
    "The implicit equation of an ellipse is \n",
    "\n",
    "$$\n",
    "\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\n",
    "$$\n",
    "\n",
    "where $a \\ne 0$ and $b \\ne 0$. The parametric representation of a ellipse is\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    x(s) &= a \\cos s\n",
    "    \\\\\n",
    "    y(s) &= b \\sin s\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "We can plot the parametric function with SymPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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AYmKAqCjVaYjoTpw8KafGtWkDvPSS6jTuZZtWRnY1a0pv6uhRoH591WnI1djK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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe48f423358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fe48ef55c18>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "from IPython.display import display\n",
    "from sympy.plotting import plot_parametric\n",
    "\n",
    "# This command makes plots appear inside the browser window\n",
    "%matplotlib inline\n",
    "\n",
    "# Create independent variable and function for x and y\n",
    "s = Symbol('s')\n",
    "xs = Function('xs')\n",
    "ys = Function('ys')\n",
    "\n",
    "# Pick concrete values for a and b\n",
    "a = 5\n",
    "b = 3\n",
    "\n",
    "# Parametric representation of ellipse\n",
    "xs = a*cos(s)\n",
    "ys = b*sin(s)\n",
    "\n",
    "# Plot parametric line\n",
    "plot_parametric(xs, ys)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tangent to the line\n",
    "\n",
    "The tangent vector $\\boldsymbol{t}$ to the surface is given by\n",
    "\n",
    "$$\n",
    "\\boldsymbol{t} = \\frac{d x}{d s} \\boldsymbol{e}_{1}\n",
    " + \\frac{d y}{d s} \\boldsymbol{e}_{2}\n",
    "$$\n",
    "\n",
    "We can compute the tangent vector via differentiation, and plot the tangent at a given point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t = (diff(xs, s), diff(ys, s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The normal direction to the surface is given by $\\nabla f$ \n",
    "\n",
    "$$\n",
    "\\nabla f = \\frac{\\partial f}{\\partial x} \\boldsymbol{e}_{1} + \\frac{\\partial f}{\\partial y} \\boldsymbol{e}_{2}\n",
    "$$\n",
    "\n",
    "evaluated at a point on the surface."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAvBAMAAAAfs4k/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACX0lEQVRIDb2Wv2sTYRzGn8tdLrlLLzkquElD\nK261wcHi5A2KU/VwcBIaFHQKFv8ACRVE6BI7aOPUycFBUl0t7eRWGq2L1KGKIE7tiSKIEC953/dy\n73vfN1KEvsv3x/N53l9J7gKw0Vj3eaYErcA4OzBbiuMfAqeLwdgP0qgVOF0OrZ+kUSsktBslqZxo\nBY4VqzKfVFqBE42EVBKtwLhCV+FFqRU4cA3XBSpHrcCwsafP12QDr7QCMD518VS13OsdqMbj72B9\nogQGGvXCYpH81tRfIJeZbTi77duRGwzrJLNWP6DUScpMYsChVQOfUalm+FRDp+YjbG+muEwaq36m\nGTe8Fk5Tfd5z/RnkSGOphlcjjPOrG7hE6l4n/4sUWPPY5PjtBVI3zk7Tv1GSlprxKenhkGcT7DlM\nhCKXo7VBGY2AU4t4KfOiMtbPU0a3yoGr05sCVePMSKNKp+ojMdrtwegO1v2vFXvDwQ4hLqeiCiPP\nGG9o5Vu7zTaUIpX0MFu1Hw5GbTDFYYzSmkdu3Nm/vwDM7b4Glv2tkG9G3Gq/PHHjvbTFdGHUcCfE\nfu+7aObjyfgwvuJyIAo1mj7Ke3h8hgLMPZgd1SDqchPmH9REKcVKC7butQkn0hsnOsjRL2o2fzHC\n7N2P0lqsKMcrjnpezdexhith1mkfwEkuLStjqd8rNQllGc9GrOgNLs6jHoWFyTe/ifl4awqI/x3Z\n9C2Y2luF28WF+BY8mnDoD6q/5hZw0gOol2vhC7YDvq9MsB7c2mlaXdwjiNwj40nGIBpO/IRoYu7m\nW9FIx8Yu8SEx4C+Ft5lmzBpP4AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1$$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x    y     \n",
       "── + ── = 1\n",
       "25   9     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAyBAMAAACt5eg9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nREwn7BZM8734C5r83IrumTBlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( \\frac{2 x}{25}, \\quad \\frac{2 y}{9}\\right )$$"
      ],
      "text/plain": [
       "⎛2⋅x  2⋅y⎞\n",
       "⎜───, ───⎟\n",
       "⎝ 25   9 ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = symbols('x y')\n",
    "f = Eq(x**2/a**2 + y**2/b**2, 1)\n",
    "display(f)\n",
    "grad_f = (diff(f.args[0], x), diff(f.args[0], y))\n",
    "display(grad_f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now plot the ellipse, and the tangent vector $\\boldsymbol{t}$ and the normal $\\boldsymbol{n} = \\nabla f$ to the surface at a point $(x(s_{0}), y(s_{0}))$. For later visualisaton, we will scale the vectors dependent on the size of the the ellipse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Pick point to evaluate tangent and normal vector at\n",
    "s0 = 0.4\n",
    "x0, y0 = xs.subs(s, s0), ys.subs(s, s0)\n",
    "\n",
    "# Evaluate tangent and normal vector, and scale according to ellipse major axis length \n",
    "t0 = (t[0].subs(s, s0), t[1].subs(s, s0))\n",
    "l = sqrt(t0[0]**2 + t0[1]**2)/max(a, b)\n",
    "t0 = (t0[0]/l, t0[1]/l)\n",
    "\n",
    "n0 = (grad_f[0].subs(x, x0).subs(y, y0), grad_f[1].subs(x, x0).subs(y, y0))\n",
    "l = sqrt(n0[0]**2 + n0[1]**2)/max(a, b)\n",
    "n0 = (n0[0]/l, n0[1]/l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now plot the ellipse, as well as the tangent and normal vectors at the point $(x(s_{0}), y(s_{0}))$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vs4qCPXtsDcDnn4dvv1XyFgnCeedZ9yVAuXK2mMT6262oSud3OgcYGTByZH7ynjatWMk7\nVErgRbB1q82iXLXK6pgce2zQEYmkrjp1rDAc2EXNnyYfwcsXvMzIhSOZsHRC7APyHtq0gfPPt1mV\n2dnQokVUD6kEHqK1a63GTK1a8PHHWkRYJB6ULm15s1EjW8Wq3Lw/c1jZwzjj1TPYk70ndoHMnWt1\nTKZMsSWI5s+3n6NMCTwEP/9so0vOO89Wgk9LCzoiESlo7lwbL3711XBr9joAajxRIzYH793bSiqC\nrfB82WWxOS5K4Ac1f76NP/3nP2HQoPBX2haR6PjwQytTe/ddJbl0yQo27tjIkGlDonfAjRstITz7\nrC035H3MT801CuUA5s+3WZWDBtnQJRGJf089ZQ2uuucOZ+kpF7L+9vUcUT7CRfeHDs1PCosX29T4\nCAtlFIo6A/Zj3jxL3v/+t5W5FJHE0LevNYz79u0CmY9RhSqRm6W5Z4/N6d+wATp2tBEnAZ6Wqwul\nEIsX2/DABx9U8hZJRP/4Bzz6KPDdbTD2AW4eeXP4b/rNN3bVdMMGm8U3alTgfarqQtnL2rV2wfKf\n/7QCOiKSuB54AO65Bzj/78we2ocmVZsU/U28hy5drFpdxYq2GnkMVh6PaT3wAwSRMAk8M9OGCnbq\nZP3eIpL4evSwLmu6dyDntTFFm6W5dCkcfbTdf/FFWxMxRpTAi2DPHujcGerWhRdeCPzMSEQiqEnT\nbObOKUmzQZcx855hob3o/vttdAlYq7tKlegFWAgl8CLo3dv6vj/5ROO8RZJRXqPsf1NH07Vlx/3v\nuHVr/nDAnj2t8FEAlMBD9Mor8Mgjtg5f3vJ0IpJcsrKgVCm7v233dsqXKr/vTh99BJfkru4zcyY0\naxa7APeiYlYhmDzZFmP4+GMlb5FklpYGS5bakj4VGkz645M5OdC0qSXvli3t5wCTd6hSOoFv2GCz\nXocMgcaNg45GRKKtbp0SPPriElhyNpf3H2UP/vijrTo+Zw4MHw5TpybMRbCU7ULxHi691CZQPfFE\n0NGISCzVPXk6y75vweLL/ka9YS/ag9u25ZeBjQPqAz+AIUPgmWdsjdEyZYKORkRiLa+R7R9+xPpR\n40ysVqV/xTm3xjk3M9z3ipWFC6FfP1uAWMlbJDVNmDQfgAbjjw84kuKLRB/4UOAAY3Lii/dw441w\n9912zUJEUtNpJzekY48pLBzZiY8nzQg6nGKJSBeKc+5o4FPv/T5fZfHWhfLqq/DcczBpkl23EJHU\nlteVsic7i7QS8TMJRMMI97J+vXWdvPCCkreImDnzbV22at3uDjiSoovJ183AgQN/v5+enk56enos\nDruPfv2gWzcb5ikiAtC4QRmOb/0bM99/mNenvc21La8KJI6MjAwyMjKK9JqU6UKZNcvqey9YAJUq\nBR2NiMST7OzcEhr1xvLbnJOoXDb4WX3qQingrrusBa7kLSJ7K1kSHnzQwy/tOWxQraDDCVkkhhG+\nA0wEGjjnljvnbgg/rMiaOBFmzFB9bxHZvzvvzG3svvIt/xrzr2CDCVFKTOQ5+2y45hqtaykiB/b8\n89CrF3DHYSy4/XvqH1E/sFg0ExMra3DRRVYqNq8SmYjI/jgHVJ8BPVuQMyCnaAtARDQO9YHz5JPQ\np4+St4iEZuBAYM2JkOM467Wzgg7ngJK6Bb58OTRvbq1vlYoVkVB4DyVKwPnXzWFEvaaM6T6GDsd0\niHkcKd8Cf/FF6/tW8haRUDkH7drBiNeaUKdSHc554xx2Ze0KOqxCJW0LPCcHjjnGFmo48cSYH15E\nEtj69VC1KkyanE3bUWmUKlGK3ffsjmkMKd0C/+YbW9auefOgIxGRRJO3fvHlXUsy+cbJ7MnZw7Pf\nPxtsUIVI2gT++uvQvXvCLKwhInHm1lth2TJoU7MNVzS9gt6jerNm65qgw/qDpOxCyc6GatVs8k7t\n2jE9tIgkiZ07oVw5GDMGOnQAd5+1Bv29sclnKduFMmUK1Kyp5C0ixVe2rN3+7W92u6LvCgBu+CR+\nJpsnZQIfNQo6JswSEyISry65xIYhA9SsWJPHznmMV2e8yvTV04MNLFdSdqGcfDI89JBNoRcRKa55\n86Bx4z+ud5zXlZI9IJsSLnpt4JTsQtm2DWbOhFNPDToSEUl0jRrZ7XPP5T+2vf92AJo81ySAiP4o\n6RL4jBnQrJkWKxaRyHnmmfz75UqVY8RVI5i/YT7D5gwLLiiSMIH/8AO0ahV0FCKSLNq1gxUr/vhY\np/qdaH1Ua7q+35XMXZnBBIYSuIjIAd2wn0Enk2+cDEDFhyvGMJo/SroEPncuNG0adBQikiwuv9xu\nt2//4+POOeb0mgPAwIyBsQ0qV9Il8CVLoF69oKMQkWSRtwzjxIn7Pte4amN6t+nNfePuY8mmJTGN\nC5IsgWdm2rdk1apBRyIiyWby5MIff+Y8u8JZ7z/1iPWQ6aRK4EuXQt26qn8iIpE3adL+n9vwrw0A\ndHq7U4yiMZFY1Lijc26ec26hc+6OSARVXBs2WA0UEZFI++GH/T93eLnDGdJlCKMXjWb80vExiyms\nBO6cKwk8C3QEmgBXOucaRyKw4ti6FQ45JKiji0gy+/XXAz/fo0UPjih3BGe+eia7s2NTOzzcFngb\nYJH3fon3fg/wLnBh+GEVjxK4iERD3oXMg1l962oAqj9ePYrR5As3gdcElhf4eUXuY4HYuhUqVAjq\n6CKSrEJdlrFUyVJMuGECm3Zu4qWpL0U3KCAtzNeHdMl14MCBv99PT08nPT09zMMWrkQJW5BURCSS\njj/eBkmE4rQ6p9G5QWf++tlfubjxxVQpXyWk12VkZJCRkVGkuMKqRuicawsM9N53zP35TiDHe/9I\ngX1iVo3wnXdg+HC7FRGJlKeftjkmTz8d2v7ee0oMsg6O4i4AEYtqhD8A9Z1zRzvnSgNXAMPDfM9i\nK1vWVtEQEYmkLVtsjd1QOedYcssSAHqN6BWdoAgzgXvvs4Cbgc+BOcB73vu5kQisOMqW3Xe6q4hI\nuLZsgYpFLHlSt3JdBqUP4vkfnmf22tlRiSvsceDe+1He+4be++O89w9FIqjiql4d1q4NMgIRSUZF\nbYHnuefMewBo9nwzcnxOhKNKspmYtWrtW/ZRRCRcK1faOrvFkXmnlZtt/VLrCEZkkiqBV6li9VDU\nDy4ikfTLL8UvkndI6UP48PIPmbZ6Gp/O/zSicSVVAi9Rwr4lly0LOhIRSRY5OeElcICLG19M06pN\n6fJuF7bvidyFuqRK4GC1wGfNCjoKEUkWq1fbTMxwJwn+1PMnACo8GLnZhkmXwE880dbFFBGJhOnT\noXnz8N+nhCvBjL9Zcnrkm0cOsneI7xmRd4kjSuAiEknffw9t2kTmvZrXaM6fW/yZfmP7sWJL+CMu\nki6Bt2hhZR81pV5EIiGSCRzg5S4vA1D7qdphv1fSJfCjj4ZSpWDevKAjEZFEl51tCbx1hEcArrlt\nDQBd/9c1rPdJugTuHJxzDnz5ZdCRiEiimzLF5pfUqBHZ961WoRrPdXqOYXOHMWnFAZb6OYikS+AA\nHTrAmDFBRyEiiW7ECOgUpVXSerXuRdm0srQb0o6snKxivUfSJvDx41UXRUTCM3IknH9+9N5/0x2b\nAKj7dN1ivT4pE3iVKtZnNWJE0JGISKJavtxKyLZrF71jlEkrw9hrx7IqcxVv/PhGkV+flAkc4Mor\nVRdcRIrvjTfg8sshLdxlbw7i7Hpnc/bRZ3Ptx9fy247fivTasBZ0COkAMVzQoaBNm6BuXZtWH+p6\ndiIiYMOQGzWC116Dtm1jcbx9F4CIxYIOcatyZTj3XHjrraAjEZFEMyl3YMjJJ8fmeM45FvVeBMCt\nn98a8uuSNoED3HwzDB5sxWhEREL14ovQo4cNS46VYw8/ln6n9uPJSU+yYMOCkF6TtF0oYKdBJ54I\njz1mrXERkYNZudIWMV60CA4/PPbHd/flfmsMJHW7UMC+Pfv0CX0hUhGRwYOhe/dgkjfkDy0MRVK3\nwAF27IDjjrPV6k86KbAwRCQBZGZa3e8ffrCyHEF5d9a7XHn8ldFrgTvnujrnZjvnsp1zLYv7PtFW\nrhz07w8DBgQdiYjEu6eego4dg03eAN2adQtpv2K3wJ1zjYAc4AXgVu/9tP3sF2gLHGDXLmjQwMaF\nn3JKoKGISJxauxaaNLH6J+GsvhMpUR1G6L2f570P7VJpwMqUsRb4XXepzKyIFO6BB+Caa+IjeYcq\nqS9iFnTddbBxI7z/ftCRiEi8WbgQ3n7bGnmJ5ICTRJ1zY4DCCin2996HvLzywIEDf7+fnp5Oenp6\nqC+NmLQ0ePZZuOoqOO88OPTQmIcgInHIe+jVC/r1g6pVg4sjIyODjIyMIr0m7FEozrmvifM+8IKu\nuw6qV4dHHw06EhGJB2++CU88YX3f0a57UhSxnEofw/lK4Xn0UXj1VZhW6NeNiKSSDRvgttts5mU8\nJe9QhTOM8GLn3HKgLTDCOTcqcmFFT/XqNlSoe3fYuTPoaEQkSLfcYhUHI71kWqwk/USewngPV1xh\nSyU9+WTQ0YhIEN58E/79b5g6FcqXDzqafYXShZKSCRzs1OmEE6zm79lnBx2NiMTS4sVWaXDMGKuX\nFI9SupzswRxxBAwdal0pq1YFHY2IxMqePXD11TZkMF6Td6hStgWe54EHYNQo+PprKF066GhEJNp6\n94ZffrH6SCXiuAmrLpQQ5OTARRfZ6j2DBwcdjYhE08svw+OPw+TJ8b9SlxJ4iDZtsqvQd95pRdxF\nJPlMnGiNtQkToGHDoKM5uFASeAKOfIy8ypXhs8/gzDOhZk3405+CjkhEImnxYrjsMlvjMhGSd6ji\nuAcotho2hA8+sIuamuQjkjzWrLEVue65x8poJBMl8AJOPRX+7//gggvsIoeIJLYtWyxpd+8OPXsG\nHU3kqQtlL5dcAr/+Cu3bQ0YG1KkTdEQiUhw7dlifd7t2ybugixJ4IXr1gt27bYJPRobN2BSRxLF9\nuyXvGjXgmWdiu7p8LCmB78c//gHZ2XDWWTBuHBx1VNARiUgotm2DLl3sb3boUChZMuiIokcJ/ABu\nvRWysmx0yhdfJNZKHSKpaOtW6NzZ1rQcMiS5kzcogR/UHXfAIYfA6afDyJFWP0VE4s+6dTYAoVkz\nKw8bz7MsIyUFPmL4brrJCr6fc45NAhCR+PLzzzaKrH17eOml1EjeoAQesiuusPKTl15q48VFJD78\n8IOdIffta+Vhk/WCZWE0lb6Ipk2zq9s33mgTA1Lpl0Uk3nz0Efz1r9bqvuiioKOJLNVCiZLVq+Hi\ni60A1tCh8VkMXiSZ5eTAfffZ398HHyTuijoHonrgUXLkkTY+vEwZO3VbsiToiERSx+bN1tr+6itb\niDgZk3eolMCLqWxZK4xz9dW2ssfw4UFHJJL8Zs60v7fatWHsWFvjNpWF1YXinHsM6AzsBn4GbvDe\nb95rn6TrQtnbd99Bt25W7eyhh7QwhEikeW91igYMsHre110XdETRF/U+cOfcOcBY732Oc+5hAO99\nv732SfoEDrBxo/1SrVsHb78NxxwTdEQiyWHjRvjLX6zA3DvvJFc52AOJeh+4936M9z4n98fJQMpW\nDTn8cOtGueIKO8V74QVrNYhI8X39NbRoYV0m332XOsk7VBEbheKc+xR4x3v/9l6Pp0QLvKA5c+Da\na6FKFZvOW7Nm0BGJJJbMTJsF/emn1hjq1CnoiGIvIivyOOfGADUKeaq/9/7T3H3uAnbvnbzzDBw4\n8Pf76enppKenH+ywCa1JE2stPPSQtR4efdS6VzRmXOTgxo61eRZnn20XLStXDjqi2MjIyCAjI6NI\nrwm7Be6cux74C9Dee7+zkOdTrgVe0LRp1n9XsSI8/zw0ahR0RCLxacMG6NcPPv/capl07Bh0RMGK\neh+4c64jcDtwYWHJW6BlS1sB+6KL4LTT7Cr6Tv1LifwuJ8dWi2/SxCbFzZyp5B2qcEehLARKAxtz\nH/rOe99rr31SugVe0IoVcMst8NNP8OSTVvZS3SqSyqZPtwVUnIP//hdOPDHoiOKHptLHqVGjrNb4\nUUdZlcPmzYOOSCS2Vq2Ce++1i5QPPgjXX586FQRDpan0ceq88+DHH62eyrnnWh/5r78GHZVI9GVm\nWjfi8cc7VEp/AAAKNUlEQVTb0Nt586BHDyXv4tI/W0BKlbI64/PnQ6VK0LQp9O9vkxZEks3u3TaT\nskEDm5AzbRo88kjqjDCJFiXwgFWubFODp02zWZwNGsCgQbBlS9CRiYRvzx67QNmwoVUN/OwzeOMN\nq+Qp4VMCjxN161pN40mTYNEiOO44G0e+efPBXysSb/ISd4MG8N57lrTHjIGTTgo6suSiBB5njjsO\nXn8dxo2DWbOspsqdd6qPXBLDtm3w7LP7Ju7TTgs6suSkBB6nGjeGt96y5aIyM22M7N//bq1zkXiz\nZo2tUHX00Van+623lLhjQQk8ztWrZy2aefOgalVo2xa6dLE/Do3OlKDNmmVLmjVqBOvXw8SJ8OGH\ncMopQUeWGjQOPMFs326tm8GD7cr+zTdb4ayKFYOOTFLF7t12QfL55201+L/+1SbjVK0adGTJRRN5\nkpj3MGGCJfKxY6FrVxtP26aNZndKdCxZYhfahwyxYa+9etnZYKlSQUeWnJTAU8TKlba82yuv2FJv\nPXrANddAtWpBRyaJbutWa22/+qrVKLnmGrsWo6Js0acEnmK8h/HjLZF/8gmkp8NVV1nNlfLlg45O\nEkV2to2Ceu01W6Tk9NOtHHLnzraQt8SGEngK27IFhg2zoVyTJ1tB/CuvhD/9SWt2yr6ys+Hbb+H9\n9+33pkYNu7Zy1VVaODgoSuACwNq19kf57rswe7a1pC680JJ5hQpBRydBycqyUSPDhtlWrZpdS+na\n1cZxS7CUwGUfK1bAxx9bF8vkyXDGGZbML7jAWl2S3DZtgtGjbUr76NG21uSll1rS1nqT8UUJXA5o\n0yYrbfvJJ7YKyrHHWnXEc8+1cbzqakl8OTlWf37MGBg5EqZOtS/tzp1tq5Wyy5DHPyVwCdmePbaO\n5xdf2DZvnv2hn3sunHWWDRtTyc/E8Msv8OWXtn31lZVtbd/eyhi3b68L2olCCVyKbcMGG1/+xRc2\nsmX9epsWffrplthbttT433iQnW3XNb79Nn/budMSdYcOdlu7dtBRSnEogUvErF5tE4cmTLCEvngx\ntGgBrVvb5KHWrW3avyYRRdeaNdYNMnWqXYCcNMlmQJ52Gpx6qm0NG+r/IRlENYE75+4HugAe2ABc\n771fXsh+SuBJaNMmmDIlf/v+e9i1yxJ5ixa24soJJ9hoBrXUi857WLbMao3kJeypU62UQsuWVpa1\nXTu7VqEJW8kp2gn8UO99Zu793kBz7/2NheynBJ4iVq2yZD5jhl04mzkTli+3FuHxx0OzZpbQGzSw\nC6ZlywYdcfCysy1RL1hgXSGzZ1vSnjPH6ts0bZqfsE86SWc5qSRmXSjOuTuBSt77foU8pwSewrZt\ns2Q0c6Ylp4ULLVktWWLDFhs0gPr1bUGLunWhTh3batSAkiWDjj583tv1gxUrbFu82ApALVpkt0uX\nWgu6fn1L1gW3ww4LOnoJUtQTuHPu30B3YDvQ1nu/qZB9lMBlH1lZlrwWLLCkvmyZ/bxsmW0bN8JR\nR9lWvXr+Vq1a/m3lyraeaOXKcMghsRkl4z3s2GE12jdutOS8bl3+7bp1NnFq5UpL2CtX2mSp2rWh\nZk1rQR97rC3ccdxx9rPORKQwYSdw59wYoLDpHf29958W2K8f0NB7f0Mh76EELkW2a5d1v6xebRfu\n1qyxxJh3f90664ffvNlut2+3LodKlSxhlitnibFs2fz7ZcpY90NhW3a2lUndvduOXfB+ZqZtW7bY\nbVoaHHqoDc+rUsUuIlat+sf7tWrZVrOmhu1J8YSSwNMO9KT3/pwQj/U2MHJ/Tw4cOPD3++np6aSn\np4f4tpKqypTJb6WGIivLEmxeMt+507YdO/Lv79xpLejCtrQ0m7i091amjCXrQw+1L4hDD9VFWYmO\njIwMMjIyivSacC5i1vfeL8y93xto473vXsh+aoGLiBRR2C3wg3jIOdcQyAZ+BnqG8V4iIlJEmsgj\nIhKHQmmBq7qFiEiCUgIXEUlQSuAiIglKCVxEJEEpgYuIJCglcBGRBKUELiKSoJTARUQSlBK4iEiC\nUgIXEUlQSuAiIglKCVxEJEEpgYuIJCglcBGRBKUELiKSoJTARUQSlBK4iEiCUgIXEUlQSuAiIgkq\n7ATunLvVOZfjnDs8EgGJiEhowkrgzrnawDnA0siEk3gyMjKCDiGq9PkSVzJ/Nkj+zxeKcFvgTwL/\nikQgiSrZf4n0+RJXMn82SP7PF4piJ3Dn3IXACu/9TxGMR0REQpR2oCedc2OAGoU8dRdwJ3Buwd0j\nGJeIiByE894X/UXONQPGAttzH6oFrATaeO/X7rVv0Q8gIiJ47w/YMC5WAt/nTZz7BTjJe78x7DcT\nEZGQRGocuFrZIiIxFpEWuIiIxF7MZmI653o75+Y652Y55x6J1XFjKRknNTnnHsv9f/vROfehc65S\n0DFFgnOuo3NunnNuoXPujqDjiSTnXG3n3NfOudm5f299go4pGpxzJZ1z051znwYdSyQ55yo754bl\n/t3Ncc613d++MUngzrmzgC7ACd77ZsDjsThuLCXxpKYvgKbe++bAAmz0UUJzzpUEngU6Ak2AK51z\njYONKqL2AH29902BtsBNSfb58twCzCH5unD/A4z03jcGTgDm7m/HWLXAewIPee/3AHjv18XouLGU\nlJOavPdjvPc5uT9OxkYcJbo2wCLv/ZLc38l3gQsDjilivPe/eu9n5N7fiiWAo4KNKrKcc7WATsDL\nJNEQ5twz3NO9968AeO+zvPeb97d/rBJ4feAM59wk51yGc65VjI4bEyk0qakHMDLoICKgJrC8wM8r\nch9LOs65o4EW2JdvMnkKuB3IOdiOCaYesM45N9Q5N80595Jzrvz+dj7gRJ6iOMiknzTgMO99W+dc\na+B/wDGROnYsJPOkpgN8tv7e+09z97kL2O29fzumwUVHsp1yF8o5dwgwDLgltyWeFJxznYG13vvp\nzrn0oOOJsDSgJXCz936Kc+5poB8wYH87R4T3/pz9Peec6wl8mLvflNwLfUd47zdE6vjRtr/Plzup\nqR7wo3MOrIthqnNun0lN8epA/3cAzrnrsdPV9jEJKPpWArUL/Fwba4UnDedcKeAD4E3v/cdBxxNh\npwBdnHOdgLJARefc6977awOOKxJWYGfzU3J/HoYl8ELFqgvlY+BsAOdcA6B0IiXvA/Hez/LeV/fe\n1/Pe18P+A1omSvI+GOdcR+xU9ULv/c6g44mQH4D6zrmjnXOlgSuA4QHHFDHOWhJDgDne+6eDjifS\nvPf9vfe1c//eugFfJUnyxnv/K7A8N08CdABm72//iLXAD+IV4BXn3ExgN5AU/9j7kWyn54OB0sCY\n3DOM77z3vYINKTze+yzn3M3A50BJYIj3fr9X+hPQqcA1wE/Ouem5j93pvR8dYEzRlGx/c72Bt3Ib\nFz8DN+xvR03kERFJUFpSTUQkQSmBi4gkKCVwEZEEpQQuIpKglMBFRBKUEriISIJSAhcRSVBK4CIi\nCer/AetPE8/RL1U1AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe47f5d01d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy.utilities.lambdify import lambdify\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Set plot limits (based on ellipse size)\n",
    "plt.xlim(-(a + 1), a + 1)\n",
    "plt.ylim(-(b + 1), b + 1)\n",
    "\n",
    "# Make plot aspect ratio equal -> normal lines appear normal\n",
    "ax = plt.axes()\n",
    "ax.set_aspect('equal')\n",
    "\n",
    "# Prepare the symbolix expression for x(s) and y(s) for plotting\n",
    "xs = lambdify(s, xs, modules=['numpy'])\n",
    "ys = lambdify(s, ys, modules=['numpy'])\n",
    "\n",
    "# Plot ellipse\n",
    "s = np.linspace(0, 7, 300)\n",
    "plt.plot(xs(s), ys(s))\n",
    "\n",
    "# Add tangent vector to plot\n",
    "ax.arrow(x0, y0,  float(t0[0]),  float(t0[1]), label='tangent', color='g')\n",
    "ax.arrow(x0, y0, -float(t0[0]), -float(t0[1]), color='g')\n",
    "\n",
    "# Add normal vector to splot\n",
    "ax.arrow(x0, y0,  float(n0[0]),  float(n0[1]), label='normal', color='r')\n",
    "\n",
    "# Show plot\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be seen from the plot that the gradient (red line) is normal to the ellipse."
   ]
  }
 ],
 "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": "3.5.0+"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}