{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting Bézier triangular patches with Python Plotly ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The de  Casteljau algorithm for computing   points on a triangular Bézier patch ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A Bézier triangular patch is a parameterized polynomial surface. Its  parameters belong to a triangular 2D region, $T$.\n",
    "The triangle vertices, $T_0, T_1, T_2$, are  denoted  counterclockwise like in the figure below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARsAAAEbCAYAAADqLSAhAAAABmJLR0QA/wD/AP+gvaeTAAAACXBI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\nOWzuc18AgCf/B7Q5o3KCwFvEAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='Imag/triangle.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each point $P$ in $T$ is expressed as a convex combination of the vertices $T_0, T_1, T_2$:\n",
    "\n",
    "$P=\\lambda_0T_0+\\lambda_1T_1+\\lambda_2T_2$, $\\lambda_i\\geq 0$, $i=\\overline{0,2}$, and $\\lambda_0+\\lambda_1+\\lambda_2=1$\n",
    "\n",
    "The scalars $\\lambda_i$ are the *barycentric coordinates* of  the point $P$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "If  $(x_i, y_i)$ are the cartesian coordinates of the vertices $T_i$, $i=\\overline{0,2}$, and $P(x, y)$ \n",
    "is a point in the corresponding triangular region, then its baycentric coordinates are given by \n",
    "the  solution of the linear system:\n",
    "\n",
    "$$\\left[\\begin{array}{ccc} \n",
    "x_0&x_1&x_2\\\\\n",
    "y_0&y_1&y_2\\\\\n",
    "1&1&1\\end{array}\\right]\n",
    "\\left[\\begin{array}{c}\\lambda_0\\\\\\lambda_1\\\\\\lambda_2\\end{array}\\right]=\n",
    "\\left[\\begin{array}{c} x\\\\y\\\\1\\end{array}\\right]$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a one-to-one correspondence between the points of  the triangle $T$ and the  triangle $\\Delta$, in 3D, of vertices\n",
    "$A_0(1,0,0)$, $A_1(0,1,0)$, $A_2(0,0,1)$. Namely, to each $P(x,y)\\in T$ one associates its barycentric coordinates  $(\\lambda_0, \\lambda_1, \\lambda_2)\\in\\Delta$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/triangle3D.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "A B&eacute;zier triangular patch is defined by an integer $n>0$, and a triangular net of 3D points\n",
    "${\\bf b}_{ijk}$, with $i, j, k\\geq 0$  integers such that $i+j+k=n$. Its parameterization is:\n",
    "$S: (\\lambda_0, \\lambda_1, \\lambda_2)\\in \\Delta \\to \\sum_{i+j+k=n}\\frac{n!}{i!j!k!}\\lambda_0^i\\lambda_1^j\\lambda_2^k {\\bf b}_{ijk}$    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the sum of powers of $\\lambda_0, \\lambda_1, \\lambda_2$ is $n$, the patch is said to have the total degree $n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A point on  such a surface corresponding to the parameters $(\\lambda_0, \\lambda_1, \\lambda_2)$ is computed recursively  by the *triangular de Casteljau algorithm*:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "$$\\begin{array}{lll}\n",
    "{\\bf b}_{ijk}^0&=&{\\bf b}_{ijk}\\\\\n",
    "  {\\bf b}_{ijk}^r&=&\\lambda_0{\\bf b}_{i+1\\,j\\,k}^{r-1}+\\lambda_1{\\bf b}_{i\\, j+1\\, k}^{r-1}\n",
    "+\\lambda_2{\\bf b}_{i\\,j\\,k+1}^{r-1},\\end{array}$$ \n",
    "where $r=\\overline{1,n}$, and $i+j+k=n-r$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The last computed point,  ${\\bf }b_{000}^n$, is a point on the surface."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Details on the triangular B&eacute;zier patches can be found in:\n",
    "    \n",
    "   1. G Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Morgan Kaufmann, 2002.\n",
    "   2. J. Gallier, Curves and Surfaces in Geometric Modeling: Theory and Algorithms, Morgan Kaufmann, 1999. \n",
    "    A free electronic version can be downloaded [here](http://www.cis.upenn.edu/~jean/gbooks/geom1.html).    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " The $(n+1)(n+2)/2$ control points, ${\\bf b}_{ ijk}$, are placed in a triangular  structure  in the same way as\n",
    " the points $(i/n, j/n, k/n)$   are located in the triangle $\\Delta$ of vertices  $A_0(1,0,0)$, $A_1(0,1,0)$, $A_2(0,0,1)$. \n",
    " \n",
    "We illustrate here the case  $n=3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DjC7OVRO3zvOSz9WQfJg1Pf+xNzQbFfS6zOyrmX0zs4N3AKLVKZv20L3c1yvl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\nl3EAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/ijk.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hence to generate a B&eacute;zier triangular patch of total degree $3$ we start with a net of points:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{array}{lllllll}\n",
    "   & &  &          {\\bf b}_{300} &        &  &  \\\\\n",
    "   & & & & & & \\\\\n",
    "   &  & {\\bf b}_{210} &           & {\\bf b}_{201} &  &  \\\\\n",
    "    & & & & & & \\\\\n",
    "   & {\\bf b}_{120} &  &      {\\bf b}_{111}    &  & {\\bf b}_{102} &  \\\\\n",
    "    & & & & & & \\\\\n",
    "  {\\bf b}_{030} &  & {\\bf b}_{021}&          & {\\bf b}_{012} &  & {\\bf b}_{003}\n",
    "\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The point indices in a triangular net or the indices that define points in  a triangular grid  of $\\Delta$,  as above,\n",
    " are returned by the following function::"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def indexes(n):\n",
    "    return [(i,j, n-(i+j)) for i in range(n,-1, -1) for j in range(n-i, -1, -1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(3, 0, 0), (2, 1, 0), (2, 0, 1), (1, 2, 0), (1, 1, 1), (1, 0, 2), (0, 3, 0), (0, 2, 1), (0, 1, 2), (0, 0, 3)]\n"
     ]
    }
   ],
   "source": [
    "print indexes(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In each step, $r$,  of the de Casteljau recursive formula, one computes  succesively  the convex combinations with the same coefficients, $\\lambda_0, \\lambda_1, \\lambda_2$, of all  triplets of  nearby points\n",
    "in the net ${\\bf b}^{r-1}_{ijk}$. The relative position of points in a such triplet is illustrated  in the next cell:\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{array}{lll} & *& \\\\\n",
    "*& &*\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the following we give the nets  of points resulted from each  step $r$.\n",
    "Starting with the initial control points:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{array}{lllllll}\n",
    "   & &  &          {\\bf b}^0_{300} &        &  &  \\\\\n",
    "   & & & & & & \\\\\n",
    "   &  & {\\bf b}^0_{210} &           & {\\bf b}^0_{201} &  &  \\\\\n",
    "    & & & & & & \\\\\n",
    "   & {\\bf b}^0_{120} &  &      {\\bf b}^0_{111}    &  & {\\bf b}^0_{102} &  \\\\\n",
    "    & & & & & & \\\\\n",
    "  {\\bf b}^0_{030} &  & {\\bf b}^0_{021}&          & {\\bf b}^0_{012} &  & {\\bf b}^0_{003}\n",
    "\\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "the step $r=1$ computes the points:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{array}{lllll}\n",
    "   &  &          {\\bf b}^1_{300} &        &   \\\\\n",
    "    & & & &  \\\\\n",
    "     & {\\bf b}^1_{210} &           & {\\bf b}^1_{201} &    \\\\\n",
    "    & & & &  \\\\\n",
    "    {\\bf b}^1_{120} &  &      {\\bf b}^1_{111}    &  & {\\bf b}^1_{102} &  \n",
    "  \\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "while $r=2$, the points:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{array}{lll}\n",
    "     &          {\\bf b}^2_{300} &          \\\\\n",
    "     & &  \\\\\n",
    "      {\\bf b}^2_{210} &           & {\\bf b}^2_{201}    \n",
    "  \\end{array}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the last step, $r=3$,  we get  the point on the surface, ${\\bf b}^3_{300}$, corresponding to the parameters $(\\lambda_0, \\lambda_1,\\lambda_2)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To implement the  de Casteljau algorithm we store the control points   in a list of points, concatenating the rows of the theoretical triangular net."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the function `deCasteljau_step`  that evaluates the convex combinations for one step of the de Casteljau formula. Its arguments are\n",
    "the degree `n`, the list of points, `b`, and the triplet of barycentric coordinates, `lam`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from __future__ import division"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def deCasteljau_step(n, b, lam):\n",
    "    \n",
    "    i=0\n",
    "    j=1\n",
    "    for nr in range(1, n+1):\n",
    "        for k in range(nr):\n",
    "            b[i]=lam[0]*b[i]+lam[1]*b[j]+lam[2]*b[j+1]\n",
    "            i+=1\n",
    "            j+=1\n",
    "        j+=1\n",
    "    return b[:-(n+1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "de  Casteljau algorithm is implemented by the following recursive function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def deCasteljau(n,b,lam):\n",
    "    \n",
    "    if len(b)>1:       \n",
    "        return deCasteljau(n-1, deCasteljau_step(n, b, lam), lam)   \n",
    "    else: \n",
    "        return b[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A discretization method of a triangular B&eacute;zier patch for plotting with Plotly ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " In the following we present a method of discretization of a triangular B&eacute;zier patch, such that to get the data  that define a `plotly Surface`.\n",
    " \n",
    " The basic idea is to define a rectangular grid  that covers the triangle $\\Delta$ of barycentric coordinates. The `deCasteljau` function, evaluated at each point $(\\lambda_0, \\lambda_1, \\lambda_2)$ of this grid, defines  a triangular net of points on the surface. These points are then integrated into a meshgrid over the rectangle \n",
    " `[min(X), max(X)], [min(Y), max(Y)]`, where `X` and `Y` are the lists  of x, respectively y-coordinates of the points on the surface.\n",
    " \n",
    "\n",
    "We start with   a triangular grid  of the triangle $\\Delta$  constructed from indexes returned by the function `indexes,` called for an even integer $2m$. \n",
    " \n",
    " We illustrate our method for a low value of $m=4$ (see the next image). \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/triangulation.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "On each side we have $2m+1$ points. Considering the points on the side $A_1A_2$ as reference points, we exclude from this grid the points that\n",
    "do not lie on perpendiculars at reference points, on $A_1A_2$. That is, we remove the points in the odd rows (parallel to $A_1A_2$).\n",
    "\n",
    "In fact we define a function `even_rows` that just generate only the points in the even rows of the above triangulation:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def even_row_pts(p):#p=2*m\n",
    "    \n",
    "    if p%2:\n",
    "        raise ValueError('p must be an even integer')\n",
    "    I=[2*k for k in range (p+1)]\n",
    "    return [(i/p,j/p, 1-(i+j)/p) for i in I[::-1] for j in range(p-i, -1, -1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and get a less dense grid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ZOJK0L+n8m3ryW7+pT79rKqej6DrZtk1xfhjIR1cGUmdZoYYWKQZGA/nYkR9IXfRuf5YV\napcbh4FcdeHgTXYJtetH24CclX40PKtb/jB8YSo/MJpkCuRnIqlf6kDqbCpUxoMBZaiM1yxuQzmn\nCnVXUk++FQNAppyfUbwv6TAUSsXIokIt+Tsa0FUlPqIo+YTKwGigTCUOpM4hofKYWqBQpT3mPemE\nWtrFBnBVSX3lyW5KhduBXTEwGijdjgoZSJ1shdqVs78AypnNkWRC5RnfQPeYaSY/3/Vurp/75BJq\npUXqbuhXA9ABoaPnnqSjXDt6kkqoJVxQAJurtFJlWVClllCLHpwA4HZhyW+sDAcgJZNQw6L0TP47\nU7aL0gC2F1qpznPblE6ibaoyMHqfZApAfirVMLdWqiQq1Fy/GwFoTo4DqaMn1JzXSwA0K7d9lai3\n/JXTUAyMBrDMRFIvl4HU0SpUBkYDWEVOvekxK9T5YjMDowFcqzKQepb6QOooFWop53YBtCeH+R6t\nV6jhOwwtUgDWNW+lGsUO5DqtV6glzT4E0K7UZyS3mlBTvxgA0pfyUzxau+VfaJEimQLY1ER+IHVy\nrVStVaglPuEQQBypbmy3klAZGA2gbiGvDJXQQOrGb/lDUy6noQDUbd7DnswAlUYrVAZGA2hSZSD1\njnM6ih5Pwwk1q8EGAPKT0oClxhJqjqO3AOQplf72RtZQw8DoqfxpKJIpgKbNW6mirqc2UqHmcOYW\nQFlSeIxS7Qk1pfUMAN0Se9+m1lv+ymmoHZIpgAiiDqSurUKlRQpACmIOpK4zoc4kDZTQqQUA3RTr\ndGYtCTXVc7UAuivG5vjWa6iVgdF7JFMACdmRH0g9bus/uHWFmkpDLQAsansG81YJtRIsLVIAktTm\nQOqNE2pqQwkAYJmwLHlf0kHTHUgbJdQQ4LEYGA0gA5VWqkY3zjfdlNqV1JPoNwWQvtCPui/pMBSE\njVi7Qm0r0wNAndq4s169QjUbvG6/8px8v+l+MsnUrCezYewwljLry2wQO4ylzIYya+w79ca4ZutL\n+zMwkFk/dhiSFDbOJ5LGH7I3f62Ra+acu/kl9Zx0z0nOSe41ffQ/b/0zbb2kXSc9CLHdd9IwekyP\nYpvNr1m4fr3oMfm4huFauXDtdqPHxDXbJrY0PwMLecNJs+gxhddr+ujr39L7vtvENXtshZw7kj9S\nKkl6Ra/9yCfspY/9qT72jdqz+5re1Z1P3tHDecXQ/xc9/amnLWpIkqSP63NP/YkuNRMP/lIv/t6L\nptejBRV8VU9/6qf01XnF0HuoO598wuLfbXDNNpPqZ+CLevGlj+iL1TuN8Sfsc3+bRt74s+fu6OH8\n7rwvn+NqOfO/SkK9cvv1AX3tdyV9vY4ANvWM3vyBO3r4/up779U7vyzp8UghXfiAvvYTi++9qydG\nkp6NEM4l79U7L1R/fkcP3/+M3nz1TT3znVgxSVyzTaT8GXhXT/z04nup5g35J6fWY4XSfVAp252T\nHsQu2SuxHS7EltKt2IOF2AbRY/Jx7S7EdRg9Jq7ZNrGl+RnoaN64vUJ17lRmO/Jl8bmkg9qy+fb2\nJJ3JV9FH4ZWKu/K3sD1JR3IuiVtE+a/fufzX81RpfT25ZutL8zNQUN4ws75zbqVjq40+9RQAcmdm\n4UTo7Un1xrYpM5ua2ay2yAAgI2Y2lq9kV2r9uq0PdSRpbCn23gFAg0LeO9caHQDXJlQzm699SIr7\naFYAiGDknJuvr67UCXBThTqULo5njbaJCgByYv4U1XxTdOU5qksTqoWjYs7vsh5J6oeKFQC6oO8e\ndZlcmfVsZsPwupQXr6tQx3rU5jAveZcmVDPrW6rniAFgTWEj6nyeNOXb+AaLv+6cO5E0rOa/6/pQ\nh+EPVpPoqNqPFarYsXyiPVJNR7cAIJb5RlRl7VRmJl3eRzqX3/U/DT8eKuS/Kwk1ZN8TXU2Qu/IJ\ndE+SQmLds1QnAwHA+sbOuf0l71/kuWqylU+sF3Ohl93yDyXtO+dO5i/5wawSm1MAChRu72cKd+KV\n9wfyea9nZtOFP7MrnysvNq0uTkqFynQ+WerUOXe38gePw/s9+TL3xDm3V/m10/nPAaB0YTn0zDl3\namaD+QbWxS2/c+5A15y3rSZXAOiykEynks7C+urFEsGWj5G2qXxVeyaqVAAdx3AUAKjJpk89BQAs\n+H+bwZjKRFc2rQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/even_rows.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    " Now to each two consecutive points placed on the same perpendicular to $A_1A_2$ one associates the mid point. Thus we construct \n",
    "additional points to the initial grid (the green points in the next image):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def additional_weights(m, weights):\n",
    "    \n",
    "    wei=map(np.array, weights)\n",
    "    new_weights=[]\n",
    "    a=0\n",
    "    for k in range(1, m+1):\n",
    "        for j in range(1,2*k):\n",
    "            new_weights.append((wei[a]+wei[a+2*k])/2)\n",
    "            a=a+1\n",
    "    return list(map(tuple, new_weights))        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The  function `grid` mixes the two  lists  of points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def grid(m, weights, new_weights):\n",
    "    \n",
    "    L=len(weights)\n",
    "    barycenters=[]\n",
    "    for  k in range(m):\n",
    "        L=2*k+1\n",
    "        enD=k**2+L \n",
    "        barycenters +=weights[k**2: enD] +new_weights[k**2: enD]\n",
    "    barycenters+=weights[m**2:]  \n",
    "    return barycenters"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the final grid of the triangle $A_0A_1A_2$ looks as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/finalgrid.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "To each point (triplet of weights) in this grid one associates a point on the B&eacute;zier patch. \n",
    "\n",
    "The following function discretizes a B&eacute;zier surface of degree `n`, control points `b`, and grid points `barycenters`, defined  above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def surface_points(n, b, barycenters):\n",
    "    \n",
    "    points=[]\n",
    "    for weight in barycenters:\n",
    "        b_aux=np.array(b)\n",
    "        points.append(deCasteljau(n, b_aux, weight))\n",
    "    return zip(*points)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to plot the discretized patch we need to construct three arays of shape (2m+1,2m+1) (or a list of lists) that contain in an appropriate position\n",
    "the x, y, respectively z-coordinates of the computed points.\n",
    "\n",
    "The  values in a list of coordinates are   placed  succesively in the red positions, while outside   the triangle, similar to the triangle $A_1A_1A_2$, we store `None` values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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HXWtr4rqmPoIkJSbevv2C/8wYO/dps0WolM5I6davjZ63L4jNgOlGmlq0bLPuQxFt87Zh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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='Imag/gridData.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `move_data_to_triangle` returns a list of lists. The values in a list of coordinates, `D`, are assigned correspondingly to  locations   placed within the formal triangle included in this list of lists:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def move_data_to_triangle(m, D):\n",
    "    \n",
    "    DD=[]\n",
    "    pos=0\n",
    "    for i in range(m):\n",
    "        L1=[None]*(2*m+1)\n",
    "        L2=[None]*(2*m+1)\n",
    "        idx=[m-i+J for J in range(2*i+1)] \n",
    "        \n",
    "        for k in range(2*i+1):\n",
    "            L1[idx[k]]=D[pos]\n",
    "            pos+=1 \n",
    "        for k in range(2*i+1):\n",
    "            L2[idx[k]]=D[pos]\n",
    "            pos+=1\n",
    "        \n",
    "        DD.append(L1)\n",
    "        DD.append(L2)\n",
    "    K=-(2*m+1)    \n",
    "    DD.append(D[K:]) \n",
    "    return DD\n",
    "   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    " `set_data_for_Surface` includes all previous functions that prepare data for a plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def set_data_for_Surface(n, b, m):\n",
    "    \n",
    "    if len(b)!=(n+1)*(n+2)/2:\n",
    "        raise ValueError('incorect number of control points')\n",
    "    weights=even_row_pts(2*m)\n",
    "    new_weights=additional_weights(m, weights)\n",
    "    grid_pts=grid(m, weights, new_weights)\n",
    "    X, Y, Z=surface_points(n, b, grid_pts)\n",
    "    ZZ=move_data_to_triangle(m, Z)\n",
    "    XX= move_data_to_triangle(m, X)\n",
    "    YY=move_data_to_triangle(m, Y)\n",
    "    return [XX, YY, ZZ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "A  colorscale   derived from  the future `viridis` [colormap](http://matplotlib.org/style_changes.html) is used to plot a `Surface`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "viridisCS=[[0.0, '#440154'],\n",
    " [0.06274509803921569, '#48186a'],\n",
    " [0.12549019607843137, '#472d7b'],\n",
    " [0.18823529411764706, '#424086'],\n",
    " [0.25098039215686274, '#3b528b'],\n",
    " [0.3137254901960784, '#33638d'],\n",
    " [0.3764705882352941, '#2c728e'],\n",
    " [0.4392156862745098, '#26828e'],\n",
    " [0.5019607843137255, '#21918c'],\n",
    " [0.5647058823529412, '#1fa088'],\n",
    " [0.6274509803921569, '#28ae80'],\n",
    " [0.6901960784313725, '#3fbc73'],\n",
    " [0.7529411764705882, '#5ec962'],\n",
    " [0.8156862745098039, '#84d44b'],\n",
    " [0.8784313725490196, '#addc30'],\n",
    " [0.9411764705882353, '#d8e219'],\n",
    " [1.0, '#fde725']]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import plotly.plotly as py\n",
    "import plotly.tools as tls\n",
    "from plotly.graph_objs import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "py.sign_in('empet', 'my_api_key')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`plot_Surface`  generates a triangular B&eacute;zier patch of degree `n` and control points `b`, over the triangular grid of $2m+1$ points on each side of the barycenter triangle $\\Delta$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_Surface(n, b, m,  plot_title): \n",
    "\n",
    "    x, y, z= set_data_for_Surface(n, b, m)\n",
    "    \n",
    "    trace = Surface(\n",
    "        z=z, \n",
    "        x=x,  \n",
    "        y=y,   \n",
    "        colorscale=viridisCS\n",
    "        )\n",
    "        \n",
    "    axis = dict(\n",
    "    showbackground=True, \n",
    "    backgroundcolor=\"rgb(230, 230,230)\",\n",
    "    gridcolor=\"rgb(255, 255, 255)\",      \n",
    "    zerolinecolor=\"rgb(255, 255, 255)\",  \n",
    "    )\n",
    "\n",
    "    layout = Layout(\n",
    "         title=plot_title, \n",
    "         width=600,\n",
    "         height=600,\n",
    "         scene=Scene(  \n",
    "         xaxis=XAxis(axis),\n",
    "         yaxis=YAxis(axis), \n",
    "         zaxis=ZAxis(axis)  \n",
    "        )\n",
    "        )\n",
    "    fig = Figure(data=[trace], layout=layout)\n",
    "    return fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we generate our first B&eacute;zier patch of degree $n=4$, over a grid corresponding to  $m=80$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n=4\n",
    "b=[[2.0, 3.0, 2.0],\n",
    " [1.5, 2.25, 3.0],\n",
    " [2.5, 2.25, 5.0],\n",
    " [1.0, 1.5, 4.0],\n",
    " [2.0, 1.5, 5.0],\n",
    " [3.0, 1.5, 3.0],\n",
    " [0.5, 0.75, 1.0],\n",
    " [1.5, 0.75, 3.0],\n",
    " [2.5, 0.75, 1.0],\n",
    " [3.5, 0.75, 4.0],\n",
    " [0.0, 0.0, 2.0],\n",
    " [1.0, 0.0, 5.0],\n",
    " [2.0, 0.0, 3.0],\n",
    " [3.0, 0.0, 1.0],\n",
    " [4.0, 0.0, 4.0]]\n",
    "m=80"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\"seamless=\"seamless\" src=\"https://plot.ly/~empet/2143.embed\" height=\"525\" width=\"100%\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig=plot_Surface(n, b, m,  'Bezier triangular patch of degree 4')\n",
    "py.iplot(fig, filename='Bezier-triangular-patch')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the control polyhedron of a triangular B&eacute;zier patch ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A B&eacute;zier triangular patch of degree $n$ and control points ${\\bf b}_{ijk}$ interpolates the corners of the triangular net formed by these points, that is, it interpolates\n",
    "the points ${\\bf b}_{n00}, {\\bf b}_{0n0}, {\\bf b}_{00n}$, and it is included in the convex hull of all its control points.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "To illustrate this property we define the  control polyhedron of the patch, i.e. the plotly trace\n",
    "we get connecting with lines the points in a triplet of nearby points."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `triangles` returns the lists of all triplets of nearby points in the triangular net ${\\bf b}_{ijk}$, $i+j+k=n$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def triangles(n, b):\n",
    "    \n",
    "    triplets=[]\n",
    "    i=0\n",
    "    j=1\n",
    "    for nr in range(1,n+1):\n",
    "        for k in range(nr):\n",
    "            triplets.append([b[i], b[j], b[j+1]])\n",
    "            i+=1\n",
    "            j+=1\n",
    "        j+=1\n",
    "    return triplets    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the function `triangle_lines` prepares  data for a `Scater3d` trace of the control polyhedron edges:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def triangle_lines(triplets):\n",
    "\n",
    "    line_pts=[]\n",
    "    for tri in triplets:\n",
    "        x, y, z=zip(*tri)\n",
    "        x=list(x)+[x[0]]\n",
    "        y=list(y)+[y[0]]\n",
    "        z=list(z)+[z[0]]\n",
    "        line_pts.append(Scatter3d(x=x, y=y, z=z,\n",
    "                              name='',\n",
    "                              mode='lines+markers', \n",
    "                              line=Line(color='#000762',\n",
    "                                        width=4),\n",
    "                              marker=Marker(\n",
    "                              symbol='dot',\n",
    "                              size=5,\n",
    "                              color='#000762')\n",
    "                                 )   \n",
    "                       )\n",
    "    return line_pts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `plot_surface_polyedron` plots both a B&eacute;zier patch and its control polyedron:\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_surface_polyhedron(n, b, m, plot_title):\n",
    "    \n",
    "    x, y, z= set_data_for_Surface(n, b, m)\n",
    "    \n",
    "    trace = Surface(\n",
    "        z=z, \n",
    "        x=x,  \n",
    "        y=y,   \n",
    "        colorscale=viridisCS\n",
    "        )\n",
    "    \n",
    "    triplets=triangles(n,b)\n",
    "    lines_pts=triangle_lines(triplets)\n",
    "    \n",
    "    axis = dict(\n",
    "    showbackground=True, \n",
    "    backgroundcolor=\"rgb(230, 230,230)\",\n",
    "    gridcolor=\"rgb(255, 255, 255)\",      \n",
    "    zerolinecolor=\"rgb(255, 255, 255)\",  \n",
    "    )\n",
    "\n",
    "    layout = Layout(\n",
    "         title=plot_title, \n",
    "         width=600,\n",
    "         height=600,\n",
    "         showlegend=False,\n",
    "         scene=Scene(  \n",
    "         xaxis=XAxis(axis),\n",
    "         yaxis=YAxis(axis), \n",
    "         zaxis=ZAxis(axis)  \n",
    "         )\n",
    "        )\n",
    "    fig = Figure(data=[trace]+lines_pts, layout=layout)\n",
    "    return fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data for a cubic (degree  $n=3$) triangular B&eacute;zier patch and its control polyhedron:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n=3\n",
    "d=[[2.5, 2.5, 1.5],\n",
    "  [1.3, 2, 3],\n",
    "  [3, 2.5, 3],\n",
    "  [1, 1, 2],\n",
    "  [2.5, 1, 4],\n",
    "  [4, 1, 1],\n",
    "  [0, 1, 1],\n",
    "  [2, 0, 2],\n",
    "  [3, -0.5, 1.5],\n",
    "  [5, 0, -0.5]]\n",
    "m=80"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\"seamless=\"seamless\" src=\"https://plot.ly/~empet/2141.embed\" height=\"525\" width=\"100%\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig=plot_surface_polyhedron(n,d, m, 'A cubic triangular Bezier patch and its control polyhedron')\n",
    "py.iplot(fig, filename='Bezier-cubic')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Triangular B&eacute;zier graphs and their contour plots###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A degree $n$ triangular B&eacute;zier patch is the graph of a function defined on a planar triangular region of vertices\n",
    "$T_0, T_1, T_2$,  if its control points have the coordinates\n",
    "${\\bf b}_{ijk}=(x_{ijk},y_{ijk}, z_{ijk})$, $i+j+k=n$, where $(x_{ijk},y_{ijk})$ are points in the triangular region $T$, of barycentric coordinates $(i/n, j/n, k/n)$, and $z_{ijk}$ are real values set by the user.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Initial data for defining such a graph are:\n",
    "  - the degree, $n$\n",
    "  - the triangle $T$ given as a list of three numpy arrays of shape (2,)\n",
    "  - a list, `b_z`, of $(n+1)(n+2)/2$ real values representing  $z$-coordinates of the control points "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cartesian_coords=lambda w, T: w[0]*T[0]+w[1]*T[1]+w[2]*T[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "calculates the cartesian coordinates of a point having barycentic coordinates $(w_0, w_1, w_2)$, with respect to a triangle $T$.\n",
    "\n",
    "The control points of the graph are returned by the following function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def control_points(n, T, b_z):\n",
    "    ind=indexes(n)\n",
    "    weights=[(i/n, j/n, k/n) for (i, j, k) in ind]\n",
    "    b_xy=[cartesian_coords(w,T) for w in weights]\n",
    "    b_xy=map(list, b_xy)\n",
    "    len_bxy=len(b_xy)\n",
    "    b=[b_xy[k]+[b_z[k]] for k in range(len_bxy)]\n",
    "    return b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us define a B&eacute;zier triangular graph of degree $n=4$, over the triangle $T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n=4\n",
    "m=80\n",
    "nr_ctrl_pts=(n+1)*(n+2)/2\n",
    "T=[np.array([0, 1.5]), np.array([-1,0]), np.array([1, 0])]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$z$-coordinates of the control points are set arbitrary in the interval $[1,5]$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b_z=1+4*np.random.random(nr_ctrl_pts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then the control points of our graph are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b=control_points(n, T, b_z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\"seamless=\"seamless\" src=\"https://plot.ly/~empet/2281.embed\" height=\"525\" width=\"100%\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig=plot_Surface(n, b, m,  'Graph as a triangular Bezier patch')\n",
    "py.iplot(fig, filename='triangular-Bezier-graph4')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Contour plot of a triangular B&eacute;zier graph ###"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having a method to generate a triangular B&eacute;zier graph, we  can generate its contour plot with Plotly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We set the data  for a plotly `Contour`, namely a 2D array or a list of lists for z-values on the surface, and two 1D lists for x, respectively y-coordinates of the grid points in the triangle $T$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n=4\n",
    "m=80"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "weights=even_row_pts(2*m)\n",
    "new_weights=additional_weights(m, weights)\n",
    "grid_pts=grid(m, weights, new_weights)\n",
    "X, Y, Z=surface_points(n, b, grid_pts)\n",
    "ZZ=move_data_to_triangle(m, Z)[::-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get the x-list for `Contour`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "K=-(2*m+1)  \n",
    "XX= [cartesian_coords(w,T)[0] for w in grid_pts[K:]]   \n",
    "x=sorted(list(set(XX)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the y-list:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "I=[k**2 for k in range(m)]\n",
    "Lam=[]\n",
    "for k in range(m):\n",
    "    Lam.append(weights[I[k]+k])\n",
    "    Lam.append(new_weights[I[k]+k])\n",
    "Lam.append(weights[-(m+1)] )   \n",
    "YY=[cartesian_coords(w,T)[1] for w in  Lam]\n",
    "y=sorted(list(set(YY)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the triangular contour:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe id=\"igraph\" scrolling=\"no\" style=\"border:none;\"seamless=\"seamless\" src=\"https://plot.ly/~empet/2283.embed\" height=\"525\" width=\"100%\"></iframe>"
      ],
      "text/plain": [
       "<plotly.tools.PlotlyDisplay object>"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "title=\"Contourplot of a triangular Bezier function\"\n",
    "data = Data([\n",
    "    Contour(\n",
    "        z=ZZ,\n",
    "        x=x, \n",
    "        y=y,\n",
    "        autocontour=False,\n",
    "        zauto=False,  \n",
    "        contours=Contours(\n",
    "            showlines=False,    \n",
    "            start =min(Z),   \n",
    "            end=max(Z),\n",
    "            size=0.1\n",
    "        ),    \n",
    "        colorscale=viridisCS, \n",
    "    ),\n",
    "    ])\n",
    "layout = Layout(\n",
    "    title= title,  \n",
    "    font= Font(family='Georgia, serif',  color='#635F5D'),\n",
    "    showlegend=False,\n",
    "    autosize=False,\n",
    "    width=500,\n",
    "    height=500,\n",
    "    xaxis=XAxis(\n",
    "        range=[min(x), max(x)],\n",
    "        showgrid=False,\n",
    "    ),\n",
    "    yaxis=YAxis(\n",
    "        range=[min(y), max(y)],\n",
    "        showgrid=False,\n",
    "    ),\n",
    "    margin=Margin(\n",
    "        l=40,\n",
    "        r=40,\n",
    "        b=85,\n",
    "        t=100,\n",
    "        pad=0,\n",
    "        autoexpand=True\n",
    "    ),\n",
    "    )\n",
    "fig = Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Contourplot-triangular-Bezier-graph4')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another pair graph-contour plot can be seen here:\n",
    " - graph: https://plot.ly/~empet/2266/graph-as-a-triangular-bezier-patch/    \n",
    " - contour plot: https://plot.ly/~empet/2268/contourplot-of-a-triangular-bezier-function/   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    /*body {\n",
       "        background-color: #F5F5F5;\n",
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       "    div.cell{\n",
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       "        margin-left: 13% !important;\n",
       "        margin-right: auto;\n",
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       "    #notebook li { /* More space between bullet points */\n",
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       "        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": 1,
     "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()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}