hypocycloid-online.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypocycloid definition and animation ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Deriving the parametric equations of a hypocycloid ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On May 11 @fermatslibrary posted a gif file, https://twitter.com/fermatslibrary/status/862659602776805379, illustrating the motion of eight cocircular points. The Fermat's Library followers found  it so fascinating that the tweet  picked up more than 1000 likes and 800 retweets. Soon after I saw  the gif I created a similar Python Plotly  animation \n",
    "     although the tweet did not mention how it was generated. @plotlygraphs tweeted a link\n",
    "    to my [Jupyter notebook](http://nbviewer.jupyter.org/github/empet/Math/blob/master/fermat-circle-moving-online.ipynb) presenting the animation code.\n",
    "    \n",
    "How I succeeded to reproduce it  so fast?  Here I explain  the secret:\n",
    "    \n",
    " At the first sight you can think that the gif  displays  an illusory rectiliniar motion of the eight points,  but it is a real one.  I noticed that the moving   points lie on a rolling circle along another circle, and I knew that a fixed point on a rolling circle describes a curve called hypocycloid. In the particular case when the ratio of the two radii  is 2 the hypocycloid degenerates to a diameter in the base (fixed) circle.\n",
    " \n",
    " In this Jupyter notebook I deduce the parametric equations of  a hypoycyloid,  animate its construction\n",
    " and explain why when $R/r=2$ any point on the rolling circle runs a  diameter  in the base circle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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N1Ksnv6WkpLL9zDfeCJ99BlOmyHsGgz7sQEBmJRNEoK3sbJy4eJL3biXpgoHs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\nMTctMDUtMThUMTI6MDE6MDktMDQ6MDB68A4fAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='generate-hypocycloid.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We refer to the figure in the above cell to explain how we get the parameterization of the hypocycloid generated by a fixed point of a circle of center $O'_0$ and radius r, rolling without slipping along  the circle\n",
    "of center O and radius $R>r$.\n",
    "\n",
    "Suppose that initially the hypocycloid generating point, $P$, is located at $(R,0)$.\n",
    "After the small circle was rolling along the greater circle a length corresponding to an angle of measure, $t$, it reaches the point $P'$ on the circle $C(O'_t, r)$.\n",
    "\n",
    "Rolling without slipping means that the length the arc $\\stackrel{\\frown}{PQ}$ of the greater circle equals the length of the arc $\\stackrel{\\frown}{P'Q}$ on the smaller one, i.e $Rt=r\\omega$, where $\\omega$ is the measure of the non-oriented angle $\\widehat{P'O'_tQ}$ (i.e. we consider $\\omega>0$) . Thus $\\omega=(R/r)t$\n",
    "\n",
    "The center $O'_t$ has the coordinates $x=(R-r)\\cos(t), (R-r)\\sin(t)$. The clockwise parameterization of the circle  $C(O'_t,r)$ with respect to the coordinate system $x'O'_ty'$ is as follows:\n",
    "\n",
    "$$\\begin{array}{llr}\n",
    "x'(\\tau)&=&r\\cos(\\tau)\\\\\n",
    "y'(\\tau)&=&-r\\sin(\\tau),\n",
    "\\end{array}$$\n",
    " $\\tau\\in[0,2\\pi]$.\n",
    " \n",
    "Hence the point $P'$ on the hypocycloid has the coordinates:\n",
    "$x'=r\\cos(\\omega-t), y'=-r\\sin(\\omega-t)$, and with respect to $xOy$, the coordinates:\n",
    "\n",
    "$x=(R-r)\\cos(t)+r\\cos(\\omega-t), y=(R-r)\\sin(t)-r\\sin(\\omega-t)$.\n",
    "\n",
    "Replacing $\\omega=(R/r)t$ we get the parameterization of the hypocycloid generated by the initial  point $P$:\n",
    "\n",
    "$$\\begin{array}{lll}\n",
    "x(t)&=&(R-r)\\cos(t)+r\\cos(t(R-r)/r)\\\\\n",
    "y(t)&=&(R-r)\\sin(t)-r\\sin(t(R-r)/r), \\quad t\\in[0,2\\pi]\n",
    "\\end{array}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If  $R/r=2$ the parametric equations of the corresponding hypocycloid are:\n",
    "\n",
    "$$\\begin{array}{lll}\n",
    "x(t)&=&2r\\cos(t)\\\\\n",
    "y(t)&=&0\n",
    "\\end{array}$$\n",
    "\n",
    "i.e. the moving point $P$ runs  the diameter $y=0$,   from the position $(R=2r, 0)$  to $(-R,0)$  when $t\\in[0,\\pi]$,\n",
    "and back to $(R,0)$, for $t\\in[\\pi, 2\\pi]$.  \n",
    "\n",
    "What about the trajectory of any other   point, $A$,  on the rolling circle that at $t=0$ has the angular coordinate $\\varphi$ with respect to the center $O'_0$?  \n",
    "\n",
    "We show that it is also a diameter in the base circle, referring to the figure in the next cell that is a particularization of\n",
    "the above figure to the case $R=2r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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89hqUlGQIxQZvLxUhnw5HXL9nYSSlXty6FHKujpsaJnIP+9LOnCFfzN69tL11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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='hypocycloid-2r.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The arbitrary point $A$ on the rolling circle has, for t=0, the coordinates:\n",
    "$x=r+r\\cos(\\varphi), y=r\\sin(\\varphi)$.     \n",
    "\n",
    "The angle $\\widehat{QO'_tP'}=\\omega$ is in this case $2t$, and $\\widehat{B'O'_tP'}=t$. Since $\\widehat{A'O'_tP'}=\\varphi$, we get that the position of the fixed point on the smaller circle,  after rolling along an arc of length $r(2t-\\varphi)$,\n",
    "is $A'(x(t)=r\\cos(t)+r\\cos(t-\\varphi), y(t)=r\\sin(t)-r\\sin(t-\\varphi))$, with $\\varphi$ constant, and $t$ variable in the interval  $[\\varphi, 2\\pi+\\varphi]$.\n",
    "\n",
    "Let us show that $y(t)/x(t)=$constant for all $t$, i.e. the generating point of the hypocycloid lies on a segment of line (diameter in the base circle):\n",
    "\n",
    "$$\\displaystyle\\frac{y(t)}{x(t)}=\\frac{r\\sin(t)-r\\sin(t-\\varphi)}{r\\cos(t)+r\\cos(t-\\varphi)}=\\left\\{\\begin{array}{ll}\\tan(\\varphi/2)& \\mbox{if}\\:\\: t=\\varphi/2\\\\\n",
    "\\displaystyle\\frac{2\\cos(t-\\varphi/2)\\sin(\\varphi/2)}{2\\cos(t-\\varphi/2)\\cos(\\varphi/2)}=\\tan(\\varphi/2)& \\mbox{if}\\:\\: t\\neq\\varphi/2 \\end{array}\\right.$$\n",
    "\n",
    "Hence the @fermatslibrary animation, illustrated by a Python Plotly code in my [Jupyter notebook](http://nbviewer.jupyter.org/github/empet/Math/blob/master/fermat-circle-moving-online.ipynb), displays the motion of the eight points placed on the rolling\n",
    "circle of radius $r=R/2$, along the corresponding diameters in the base circle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Animating the hypocycloid generation ###"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import pi, cos, sin\n",
    "import copy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import plotly.plotly as py\n",
    "from plotly.grid_objs import Grid, Column\n",
    "import time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the layout of the plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "axis=dict(showline=False,  \n",
    "          zeroline=False,\n",
    "          showgrid=False,\n",
    "          showticklabels=False,\n",
    "          range=[-1.1,1.1],\n",
    "          autorange=False,\n",
    "          title='' \n",
    "          )\n",
    "\n",
    "layout=dict(title='',\n",
    "            font=dict(family='Balto'),\n",
    "            autosize=False,\n",
    "            width=600,\n",
    "            height=600,\n",
    "            showlegend=False,\n",
    "            xaxis=dict(axis),\n",
    "            yaxis=dict(axis),\n",
    "            hovermode='closest',\n",
    "            shapes=[],\n",
    "            updatemenus=[dict(type='buttons',\n",
    "                              showactive=False,\n",
    "                              y=1,\n",
    "                              x=1.2,\n",
    "                              xanchor='right',\n",
    "                              yanchor='top',\n",
    "                              pad=dict(l=10),\n",
    "                              buttons=[dict(label='Play',\n",
    "                                            method='animate',\n",
    "                                            args=[None, dict(frame=dict(duration=90, redraw=False), \n",
    "                                            transition=dict(duration=0),\n",
    "                                            fromcurrent=True,\n",
    "                                            mode='immediate'\n",
    "                                                )]\n",
    "                                            )]\n",
    "                            )]\n",
    "           )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the base circle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "layout['shapes'].append(dict(type= 'circle',\n",
    "                         layer= 'below',\n",
    "                         xref= 'x',\n",
    "                         yref='y',\n",
    "                         fillcolor= 'rgba(245,245,245, 0.95)',\n",
    "                         x0= -1.005,\n",
    "                         y0= -1.005,\n",
    "                         x1= 1.005,\n",
    "                         y1= 1.005,\n",
    "                         line= dict(color= 'rgb(40,40,40)', width=2\n",
    "                            )\n",
    "                         )\n",
    "                       )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def circle(C, rad):\n",
    "    #C=center, rad=radius\n",
    "    theta=np.linspace(0,1,100)\n",
    "    return C[0]+rad*cos(2*pi*theta), C[1]-rad*sin(2*pi*theta)\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prepare data for animation to be uploaded to Plotly cloud:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def set_my_columns(R=1.0, ratio=3):\n",
    "    #R=the radius of base circle\n",
    "    #ratio=R/r, where r=is the radius of the rolling circle\n",
    "    r=R/float(ratio)\n",
    "    xrol, yrol=circle([R-r, 0], 0)\n",
    "\n",
    "    my_columns=[Column(xrol, 'xrol'), Column(yrol, 'yrol')]\n",
    "\n",
    "    my_columns.append(Column([R-r, R], 'xrad'))  \n",
    "    my_columns.append(Column([0,0], 'yrad'))\n",
    "    my_columns.append(Column([R], 'xstart'))  \n",
    "    my_columns.append(Column([0], 'ystart'))\n",
    "   \n",
    "    a=R-r\n",
    "    b=(R-r)/float(r)\n",
    "    frames=[]\n",
    "    t=np.linspace(0,1,50)\n",
    "    xpts=[]\n",
    "    ypts=[]\n",
    "    for k in range(t.shape[0]):\n",
    "        X,Y=circle([a*cos(2*pi*t[k]), a*sin(2*pi*t[k])], r)\n",
    "        my_columns.append(Column(X, 'xrcirc{}'.format(k+1)))\n",
    "        my_columns.append(Column(Y, 'yrcirc{}'.format(k+1)))\n",
    "        #The generator point has the coordinates(xp,yp)\n",
    "        xp=a*cos(2*pi*t[k])+r*cos(2*pi*b*t[k])\n",
    "        yp=a*sin(2*pi*t[k])-r*sin(2*pi*b*t[k])\n",
    "        xpts.append(xp)\n",
    "        ypts.append(yp)\n",
    "        my_columns.append(Column([a*cos(2*pi*t[k]), xp], 'xrad{}'.format(k+1)))\n",
    "        my_columns.append(Column([a*sin(2*pi*t[k]), yp], 'yrad{}'.format(k+1)))\n",
    "        my_columns.append(Column(copy.deepcopy(xpts), 'xpt{}'.format(k+1)))\n",
    "        my_columns.append(Column(copy.deepcopy(ypts), 'ypt{}'.format(k+1)))\n",
    "       \n",
    "    return t, Grid(my_columns)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def set_data(grid):\n",
    "    \n",
    "\n",
    "    return [dict(xsrc=grid.get_column_reference('xrol'),#rolling circle\n",
    "                 ysrc= grid.get_column_reference('yrol'),\n",
    "                 mode='lines',\n",
    "                 line=dict(width=2, color='blue'),\n",
    "                 name='',\n",
    "                ),\n",
    "            dict(xsrc=grid.get_column_reference('xrad'),#radius in the rolling circle\n",
    "                 ysrc= grid.get_column_reference('yrad'),\n",
    "                 mode='markers+lines',\n",
    "                 line=dict(width=1.5, color='blue'),\n",
    "                 marker=dict(size=4, color='blue'),\n",
    "                 name=''),\n",
    "            \n",
    "            dict(xsrc=grid.get_column_reference('xstart'),#starting point  on the  hypocycloid\n",
    "                 ysrc= grid.get_column_reference('ystart'),\n",
    "                 mode='marker+lines',\n",
    "                 line=dict(width=2, color='red', shape='spline'),\n",
    "                 name='')\n",
    "           ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set data for each animation frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def set_frames(t, grid):\n",
    "   \n",
    "    return [dict(data=[dict(xsrc=grid.get_column_reference('xrcirc{}'.format(k+1)),#update rolling circ position\n",
    "                            ysrc=grid.get_column_reference('yrcirc{}'.format(k+1))\n",
    "                                     ),\n",
    "                                 dict(xsrc=grid.get_column_reference('xrad{}'.format(k+1)),#update the radius \n",
    "                                      ysrc=grid.get_column_reference('yrad{}'.format(k+1))#of generating point\n",
    "                                     ),\n",
    "                                 dict(xsrc=grid.get_column_reference('xpt{}'.format(k+1)),#update hypocycloid arc\n",
    "                                      ysrc=grid.get_column_reference('ypt{}'.format(k+1))\n",
    "                                     )\n",
    "                      ],\n",
    "                 traces=[0,1,2])  for k in range(t.shape[0])\n",
    "               ] \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Animate the generation of a hypocycloid with 3 cusps(i.e.  $R/r=3$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "u'https://plot.ly/~empet/14320/'"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "py.sign_in('empet', 'my_api_key')#access my Plotly account\n",
    "\n",
    "t, grid=set_my_columns(R=1, ratio=3)\n",
    "py.grid_ops.upload(grid, 'animdata-hypo3'+str(time.time()), auto_open=False)#upload data to Plotly cloud"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\" seamless=\"seamless\" src=\"https://plot.ly/~empet/14321.embed\" height=\"600px\" width=\"600px\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data1=set_data(grid)\n",
    "frames1=set_frames(t, grid)\n",
    "title='Hypocycloid with '+str(3)+' cusps, '+'<br>generated by a fixed point of a circle rolling inside another circle; R/r=3'\n",
    "layout.update(title=title)\n",
    "fig1=dict(data=data1, layout=layout, frames=frames1)\n",
    "py.icreate_animations(fig1, filename='anim-hypocycl3'+str(time.time()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hypocycloid with four cusps (astroid):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "u'https://plot.ly/~empet/14322/'"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t, grid=set_my_columns(R=1, ratio=4)\n",
    "py.grid_ops.upload(grid, 'animdata-hypo4'+str(time.time()), auto_open=False)#upload data to Plotly cloud"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\" seamless=\"seamless\" src=\"https://plot.ly/~empet/14323.embed\" height=\"600px\" width=\"600px\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data2=set_data(grid)\n",
    "frames2=set_frames(t, grid)\n",
    "title2='Hypocycloid with '+str(4)+' cusps, '+'<br>generated by a fixed point of a circle rolling inside another circle; R/r=4'\n",
    "layout.update(title=title2)\n",
    "fig2=dict(data=data2, layout=layout, frames=frames2)\n",
    "py.icreate_animations(fig2, filename='anim-hypocycl4'+str(time.time()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Degenerate hypocycloid (R/r=2):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "u'https://plot.ly/~empet/14326/'"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t, grid=set_my_columns(R=1, ratio=2)\n",
    "py.grid_ops.upload(grid, 'animdata-hypo2'+str(time.time()), auto_open=False)#upload data to Plotly cloud"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\" seamless=\"seamless\" src=\"https://plot.ly/~empet/14327.embed\" height=\"600px\" width=\"600px\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data3=set_data(grid)\n",
    "frames3=set_frames(t, grid)\n",
    "title3='Degenerate Hypocycloid; R/r=2'\n",
    "layout.update(title=title3)\n",
    "fig3=dict(data=data3, layout=layout, frames=frames3)\n",
    "py.icreate_animations(fig3, filename='anim-hypocycl2'+str(time.time()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>\n",
       "    /*body {\n",
       "        background-color: #F5F5F5;\n",
       "    }*/\n",
       "    div.cell{\n",
       "        width: 900px;\n",
       "        margin-left: 13% !important;\n",
       "        margin-right: auto;\n",
       "    }\n",
       "    #notebook li { /* More space between bullet points */\n",
       "    margin-top:0.8em;\n",
       "    }\n",
       "\n",
       "    h1 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\n",
       "    .text_cell_render h1 {\n",
       "        font-weight: 200;\n",
       "        font-size: 40pt;\n",
       "        line-height: 100%;\n",
       "        color: rgb(8, 66, 133);\n",
       "        margin-bottom: 0em;\n",
       "        margin-top: 0em;\n",
       "        display: block;\n",
       "    }\n",
       "    h2 {\n",
       "        font-family: 'Fenix', serif;\n",
       "        text-indent:1em;\n",
       "        text-align:center;\n",
       "    }\n",
       "    .text_cell_render h2 {\n",
       "        font-weight: 200;\n",
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       "        line-height: 100%;\n",
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       "        margin-bottom: 1.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "    h3 {\n",
       "        font-family: 'Fenix', serif;\n",
       "        %margin-top:12px;\n",
       "        %margin-bottom: 3px;\n",
       "    }\n",
       "    .text_cell_render h3 {\n",
       "        font-weight: 300;\n",
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       "        margin-bottom: 0.5em;\n",
       "        margin-top: 2em;\n",
       "        display: block;\n",
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       "    h4 {\n",
       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "    .text_cell_render h4 {\n",
       "        font-weight: 300;\n",
       "        font-size: 16pt;\n",
       "        color: rgb(8, 66, 133);\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "    h5 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\n",
       "    .text_cell_render h5 {\n",
       "        font-weight: 300;\n",
       "        font-style: normal;\n",
       "        font-size: 16pt;\n",
       "        margin-bottom: 0em;\n",
       "        margin-top: 1.5em;\n",
       "        display: block;\n",
       "        }\n",
       "    div.text_cell_render{\n",
       "        font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 145%;\n",
       "        font-size: 130%;\n",
       "        width:900px;\n",
       "        margin-left:auto;\n",
       "        margin-right:auto;\n",
       "        %text-align:justify;\n",
       "        %text-justify:inter-word;\n",
       "    }\n",
       "    \n",
       "    \n",
       "    code{\n",
       "      font-size: 78%;\n",
       "    }\n",
       "    .rendered_html code{\n",
       "        background-color: transparent;\n",
       "        white-space: inherit;   \n",
       "    }\n",
       "    .prompt{\n",
       "        display: None;\n",
       "     }\n",
       "    .rendered_html code{\n",
       "    background-color: transparent;\n",
       "    }\n",
       "\n",
       "    blockquote{\n",
       "      display:block;\n",
       "      background: #f3f3f3;\n",
       "      font-family: \"Open sans\",verdana,arial,sans-serif;\n",
       "      width:610px;\n",
       "      padding: 15px 15px 15px 15px;\n",
       "      text-align:justify;\n",
       "      text-justify:inter-word;\n",
       "      }\n",
       "      blockquote p {\n",
       "        margin-bottom: 0;\n",
       "        line-height: 125%;\n",
       "        font-size: 100%;\n",
       "      }\n",
       "   /* element.style {\n",
       "    } */\n",
       "</style>\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"]\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ [\"$\",\"$\"], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ [\"$$\",\"$$\"], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: \"center\", // Change this to \"center\" to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {\".MathJax_Display\": {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def  css_styling():\n",
    "    styles = open(\"./custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
  }
 ],
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