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   "source": [
    "## Animated construction of the Dragon curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The most known method to draw a Dragon curve is by using turtle graphics.  Here we implement a method visually illustrated \n",
    " in a video posted by [Numberphile](https://en.wikipedia.org/wiki/Numberphile): \n",
    "    [https://www.youtube.com/watch?v=NajQEiKFom4](https://www.youtube.com/watch?v=NajQEiKFom4).\n",
    "We are starting with a vertical segment and the successive rotations are counterclockwise.    "
   ]
  },
  {
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rot_matrix(alpha):\n",
    "#Define the matrix of rotation about origin with an angle of alpha radians:\n",
    "    return np.array([[np.cos(alpha), -np.sin(alpha)], \n",
    "                     [np.sin(alpha), np.cos(alpha)]])\n",
    "\n",
    "def rotate_dragon(x, y, alpha=pi/2):\n",
    "    #x,y  lists or 1D-array containng the (x, y)-coordinates of the turn points on the dragon curve constructed \n",
    "    # in a single step\n",
    "    X, Y = rot_matrix(alpha).dot(np.stack((x, y))) # the lists of coordinates of turn points on the rotated curve\n",
    "    return X, Y\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#the initial step dragon cuvre is represented by a vertical line of length L\n",
    "L = 0.12\n",
    "X = np.array([0, 0])\n",
    "Y = np.array([-L, 0])\n",
    "\n",
    "fig =  go.Figure(data=[go.Scatter(x=X,y=Y, \n",
    "                                  mode='lines', \n",
    "                                  line_color='#0000ee',\n",
    "                                  line_width=1.5,\n",
    "                                  showlegend=False)\n",
    "                    ])\n",
    "title =  \"Animated construction of the Dragon curve,<br>through successive rotations\" \n",
    "fig.update_layout(title_text=title, title_x=0.5,\n",
    "                  font=dict(family='Balto', size=16),\n",
    "                  width=700, height=700,\n",
    "                  xaxis_visible=False, \n",
    "                  yaxis_visible=False,\n",
    "                 \n",
    "                  xaxis_range=[-11, 6],\n",
    "                  yaxis_range=[-11, 3],\n",
    "                  #margin_l=40,\n",
    "                 );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The frame 0 displays the initial vertical segment, as the dragon cuve defined in step 0 of the iterative \n",
    "process of construction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = pi/10 # The rotation of 90 degrees is defined as 5 successive rotations of 18 degrees=pi10 radians\n",
    "n_rot90 = 13 # we have 13 steps\n",
    "frames = []\n",
    "\n",
    "for k in range(n_rot90):\n",
    "    #Record the last point on the dragon, defined in the previous step\n",
    "    x0, y0 = X[-1], Y[-1]\n",
    "    x = X-x0  #Translation with origin at (x0, y0)  to be the center of rotation\n",
    "    y = Y-y0\n",
    "    for j in range(5): \n",
    "        X, Y = rotate_dragon(x, y, alpha=(j+1)*alpha)\n",
    "        X = np.concatenate((x[:-1], X[::-1]), axis=None) #concatenate to the (k-1)^th step dragon  its rotated version\n",
    "        Y = np.concatenate((y[:-1], Y[::-1]), axis=None)\n",
    "        X = X+x0\n",
    "        Y = Y+y0\n",
    "        frames.append(go.Frame(data=[go.Scatter(x=X,y=Y)],\n",
    "                               traces=[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a button that triggers the animation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "buttonPlay = {'args': [None, \n",
    "                     {'frame': {'duration': 100,\n",
    "                                'redraw': False}, \n",
    "                                'transition': {'duration': 0}, \n",
    "                                'fromcurrent': True,\n",
    "                                'mode': 'immediate'}],\n",
    "                                'label': 'Play',\n",
    "                                'method': 'animate'}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig.update_layout(updatemenus=[{'buttons': [buttonPlay],\n",
    "                                'showactive': False,\n",
    "                                'type': 'buttons',\n",
    "                                'x': 1,\n",
    "                                'xanchor': 'left',\n",
    "                                'y': 1,\n",
    "                                'yanchor': 'top'\n",
    "                                }])\n",
    "\n",
    " \n",
    "\n",
    "fig.frames=frames"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"700px\"\n",
       "            height=\"700px\"\n",
       "            src=\"https://plotly.com/~empet/15635.embed\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x286fa6d7da0>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import chart_studio.plotly as py\n",
    "py.iplot(fig, filename='rot-dragon1')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A gif file derived from this animation is posted on [Wikimedia](https://commons.wikimedia.org/wiki/File:Animated_Dragon_construction.gif#/media/File:Animated_Dragon_construction.gif)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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