fermat-circle-moving-online.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An animation reproducing a Fermat's Library gif, posted on twitter ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Update:  The explanation of the rectilinear motion of the eight points in this animation is given [here](http://nbviewer.jupyter.org/github/empet/Math/blob/master/hypocycloid-online.ipynb)."
   ]
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   "metadata": {},
   "source": [
    "   Fermat's library, `@fermatslibrary`, posted on twitter  a gif [https://twitter.com/fermatslibrary/status/862659602776805379](https://twitter.com/fermatslibrary/status/862659602776805379) illustrating\n",
    " eight points moving inside a circle. No explanation of the motion was given. Here I reproduce it via Python Plotly.\n",
    " \n",
    " Following each particular point one notices that it moves on a diameter of the greater circle.\n",
    " \n",
    "The white rays intersect the  circle (of radius R=1) at the $16^{th}$ roots of  unity. The eight points lie at each time during their motion on a circle of radius 1/2, and the center at the middle of a ray of the greater circle.\n",
    "\n",
    "If  at the time t=0, corresponding to the initial frame, the smaller circle has the center $(0.5, 0)$, and radius r=1/2, then the  points at this time are represented by the complex numbers   $u[k]=0.5+0.5\\cdot e^{2\\pi j k/8}$, where $e^{2\\pi j k/8}$, $k=\\overline{0,7}$, are the $8^{th}$ roots of  unity. The $m^{th}$ frame of the animation displays  the position of  points obtained from the initial ones, as follows:\n",
    "  - rotate the center 0.5+0j about O, by $\\theta_m=-2 m \\pi/16$ and get the new center $C_m$;\n",
    "  - place  on the circle of center $C_m$ and radius r=0.5, the points $u[k]-0.5$  rotated about O with $-\\theta_m$.\n",
    "Hence in the $m^{th}$ frame we plot the points $w[k]=e^{-m\\pi j/8}0.5+e^{m\\pi j/8}(u[k]-0.5)$, $k=\\overline{0,7}$"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import pi\n",
    "import plotly.graph_objects as go "
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = np.array([0.5 + 0.5*np.exp(2*k*np.pi*1j/8) for k in range(8)], dtype=np.complex)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define data that will be updated by each animation frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig = go.Figure(go.Scatter(\n",
    "           x = u.real,  \n",
    "           y= u.imag,\n",
    "           mode='markers',\n",
    "           marker=dict( size=0.01, color='white'),\n",
    "           name='moving_pts',\n",
    "           ))"
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
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    "frames=[]\n",
    "for m in range(64):\n",
    "    w = np.exp(-1j*m*pi/8)*0.5+np.exp(1j*m*pi/8)*(u-0.5)\n",
    "    frames.append(go.Frame(data=[go.Scatter(\n",
    "                                      x= w.real,\n",
    "                                      y=w.imag, \n",
    "                                      marker=dict(size=15, color='white')    \n",
    "                                 )],\n",
    "                           traces=[0]))\n",
    "fig.update(frames=frames);                           "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the plot layout:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "axis = dict(visible=False,  \n",
    "            range=[-1.1, 1.01],\n",
    "            autorange=False)\n",
    "\n",
    "fig.update_layout(\n",
    "            title='Math is Fun',\n",
    "            width=600,\n",
    "            height=600,\n",
    "            showlegend=False,\n",
    "            xaxis=axis,\n",
    "            yaxis=axis,\n",
    "            hovermode='closest',\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=150, redraw=False), \n",
    "                                            transition=dict(duration=80),\n",
    "                                            fromcurrent=True,\n",
    "                                            mode='immediate'\n",
    "                                                )]\n",
    "                                            )]\n",
    "                            )]\n",
    "           );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The black disk is defined as a Plotly shape, and the withe diameters as quadratic B&eacute;zier curves defined by\n",
    " three colinear control points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = np.array([np.exp(2*k*np.pi*1j/16) for k in range(16)], dtype=np.complex) # the 16^th roots of unity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig.add_shape(dict(type= 'circle',\n",
    "                   layer= 'below',\n",
    "                   xref= 'x',\n",
    "                   yref='y',\n",
    "                   fillcolor= 'rgba(10,10,10, 0.9)',\n",
    "                   x0=-1.01,\n",
    "                   y0= -1.01,\n",
    "                   x1= 1.01,\n",
    "                   y1= 1.01,\n",
    "                   line_color= 'rgba(10,10,10, 0.9)'))\n",
    "\n",
    "#define the shapes for  the eight diameters\n",
    "for k in range(8):\n",
    "    x0 = z[k].real\n",
    "    y0 = z[k].imag\n",
    "    x1 = z[k+8].real\n",
    "    y1 = z[k+8].imag\n",
    "    fig.add_shape(dict(type= 'path',\n",
    "                       path= f'M{x0}, {y0} Q 0.0,  0.0 {x1}, {y1}',                     \n",
    "                       line= dict(color= 'white', \n",
    "                                  width=0.75)\n",
    "                                ))   "
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    " "
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5.551115123125783e-17, -0.3826834323650897, -0.6532814824381886, -0.6532814824381892], \"y\": [0.0, -4.440892098500626e-16, 0.2705980500730977, 0.6532814824381874, 0.9238795325112863, 0.9238795325112868, 0.6532814824381887, 0.27059805007309895], \"type\": \"scatter\"}], \"traces\": [0]}, {\"data\": [{\"marker\": {\"color\": \"white\", \"size\": 15}, \"x\": [-2.6948419387607653e-15, 0.3535533905932715, 0.49999999999999867, 0.3535533905932734, 6.123233995736757e-17, -0.3535533905932741, -0.5000000000000013, -0.3535533905932761], \"y\": [0.0, 0.14644660940672533, 0.49999999999999867, 0.8535533905932728, 1.0, 0.8535533905932748, 0.5000000000000014, 0.1464466094067272], \"type\": \"scatter\"}], \"traces\": [0]}, {\"data\": [{\"marker\": {\"color\": \"white\", \"size\": 15}, \"x\": [0.38268343236508934, 0.653281482438188, 0.6532814824381882, 0.3826834323650898, 5.551115123125783e-17, -0.2705980500730987, -0.2705980500730989, -4.996003610813204e-16], \"y\": [0.0, 0.2705980500730984, 0.6532814824381882, 0.9238795325112867, 0.923879532511287, 0.6532814824381886, 0.2705980500730989, 1.6653345369377348e-16], \"type\": \"scatter\"}], \"traces\": [0]}, {\"data\": [{\"marker\": {\"color\": \"white\", \"size\": 15}, \"x\": [0.7071067811865462, 0.8535533905932731, 0.7071067811865475, 0.35355339059327406, 5.551115123125783e-17, -0.14644660940672688, -1.3322676295501878e-15, 0.3535533905932722], \"y\": [0.0, 0.3535533905932735, 0.7071067811865475, 0.8535533905932744, 0.7071067811865488, 0.3535533905932754, 1.3322676295501878e-15, -0.1464466094067256], \"type\": \"scatter\"}], \"traces\": [0]}, {\"data\": [{\"marker\": {\"color\": \"white\", \"size\": 15}, \"x\": [0.9238795325112868, 0.9238795325112867, 0.6532814824381882, 0.27059805007309845, 0.0, 5.551115123125783e-17, 0.27059805007309856, 0.6532814824381884], \"y\": [0.0, 0.38268343236508984, 0.6532814824381882, 0.6532814824381881, 0.3826834323650895, -2.498001805406602e-16, -0.2705980500730987, -0.2705980500730987], \"type\": \"scatter\"}], \"traces\": [0]}]);\n",
       "                        }).then(function(){\n",
       "                            Plotly.animate('bb10284b-077d-4bd4-9fbc-01cb29e5e717', null);\n",
       "                        }).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('bb10284b-077d-4bd4-9fbc-01cb29e5e717');\n",
       "var x = new MutationObserver(function (mutations, observer) {{\n",
       "        var display = window.getComputedStyle(gd).display;\n",
       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\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",
       "}}\n",
       "\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",
       "}}\n",
       "\n",
       "                        })\n",
       "                };\n",
       "                });\n",
       "            </script>\n",
       "        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from plotly.offline import download_plotlyjs, init_notebook_mode,  iplot\n",
    "init_notebook_mode(connected=True)\n",
    "iplot(fig)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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",
       "        font-size: 28pt;\n",
       "        line-height: 100%;\n",
       "        color: rgb(8, 66, 133);\n",
       "        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",
       "        font-size: 18pt;\n",
       "        line-height: 100%;\n",
       "        color: rgb(8, 66, 133);\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 2em;\n",
       "        display: block;\n",
       "    }\n",
       "    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": 10,
     "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()"
   ]
  },
  {
   "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}