{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Bézier triangular patches defined as Plotly trisurfs ##" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One year ago we presented [here](http://nbviewer.jupyter.org/github/empet/Geometric-Modeling/blob/master/Bezier-triangular-patch-in-plotly.ipynb) a method for visualizing Bézier triangular surfaces via Python Plotly. Since at that time Plotly could plot only rectangular patches we devised a tricky method to associate a rectangular meshgrid to a triangulation. Now Plotly can plot trisurfs, and here we present how we triangulate the parameter domain of a triangular patch in order to plot it as a trisurf." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The new method is based on the theoretical presentation of Bézier triangular surfaces made in the previous notebook." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np\n", "from __future__ import division" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Below we define functions that perform the triangulation of the equilateral triangle $\\Delta$, in $\\mathbb{R}^3$, of vertices $(0,0,1)$, $(1,0,0)$, $(0,1,0)$. Unlike the triangulation method provided by matplotlib.tri, we define a triangulation that returns the barycentric coordinates of vertices, not the cartesian ones." ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "The triangulate function called for an integer p returns the vertices of a triangulation having $p+1$ points on each side of the triangle $\\Delta$, simplices returns the indices corresponding to vertices that form the unfilled triangles in the image below, while simplicesCompl returns the indices corresponding to vertices of filled triangles:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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I1C5AwKrdkBYQiEIg5qARY7CMYlFzkggYFCBgGcSlaQRCEog5ZMQcLkNaw5wLAnkKELDy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\nFdJsWjoXXTcmNvjaJrCWzZ0Nvjb7Wjd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"text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import Image\n", "Image(filename='Imag/triangulgroups.png')" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def triangulate(p):\n", " I=range(p, -1, -1)\n", " return [(i/p,j/p, 1-(i+j)/p) for i in I for j in range(p-i, -1, -1)]\n", "\n", "def simplices(p):\n", " \n", " triplets=[]\n", " i=0\n", " j=1\n", " for nr in range(1,p+1):\n", " for k in range(nr):\n", " triplets.append([i,j,j+1])\n", " i+=1\n", " j+=1\n", " j+=1\n", " return np.array(triplets).reshape((len(triplets), 3)) " ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def simplicesCompl(p):\n", " \n", " triplets=[]\n", " i=1\n", " j=2\n", " t=4\n", " m=2\n", " for nr in range(1,p):\n", " for k in range(nr):\n", " triplets.append([i,j,t])\n", " if k1: \n", " return deCasteljau(n-1, deCasteljau_step(n, b, lam), lam) \n", " else: \n", " return b[0]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "To each point (triplet of barycentric coordinates) in a triangulation one associates a point on the Bézier patch. \n", "\n", "The following function discretizes a Bézier surface of degree n, control points b, and triangulation vertices ( barycenters), defined above:" ] }, { "cell_type": "code", "execution_count": 8, "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": { "collapsed": true }, "source": [ " set_data_for_Surface prepare data for a plot:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def set_data_for_Surface(n, b, p):\n", " \n", " if len(b)!=(n+1)*(n+2)/2:\n", " raise ValueError('incorect number of control points')\n", " barycenters=triangulate(p)\n", " \n", " x,y,z=surface_points(n, b, barycenters )\n", " return x,y, z" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def map_z2color(zval, cmap, vmin, vmax):\n", " #map the normalized value zval to a corresponding color in the colormap cmap\n", " \n", " if vmin>=vmax:\n", " raise ValueError('incorrect relation between vmin and vmax')\n", " t=(zval-vmin)/float((vmax-vmin))#normalize val\n", " C=map(np.uint8, np.array(cmap(t)[:3])*255)\n", " return 'rgb'+str((C[0], C[1], C[2]))\n", " \n", "\n", "def tri_indices(simplices):\n", " #simplices is a numpy array defining the simplices of the triangulation\n", " #returns the lists of indices i, j, k\n", " \n", " return ([triplet[c] for triplet in simplices] for c in range(3))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "