{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Described in Goldman, used as argument that Beziers are fractals because they are fixed points of an iterated function system.\n", "\n", "The de Casteljau algorithm splits a Bezier curve into two Bezier segments.\n", "\n" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Bezier Curve Definition" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Bezier curve $P(t)$ of degree $n$\n", "\n", "$P(t) = \\sum_{j=0}^{n} B_j^n (t) P_j \\quad \\quad \\quad 0 \\leq t \\leq 1$\n", "\n", "For control points $P_0 ... P_n$, with Bernstein basis functions $B_j^n(t) = \\left(^n_j\\right)t^j(1 - t)^{n-j} \\quad \\quad j=0,...n$\n", "\n", "Binomial Coefficients: $\\left(^n_j\\right) = \\frac{n!}{n!(n-j)!}$" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Helpers" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from operator import mul # or mul=lambda x,y:x*y\n", "from fractions import Fraction\n", "\n", "def nCk(n,k): \n", " \"\"\" Binomial Coefficient nCk \"\"\"\n", " # Based on:\n", " # http://stackoverflow.com/questions/3025162/statistics-combinations-in-python\n", " # http://stackoverflow.com/questions/279561/what-is-the-python-equivalent-of-static-variables-inside-a-function\n", "\n", " \n", " # Dynamic programming is possible in python!\n", " # Not sure if it's actually worth it though\n", " try:\n", " return nCk.dynamic[(n,k)]\n", " except KeyError:\n", " nCk.dynamic.update({(n,k) : int( reduce(mul, (Fraction(n-i, i+1) for i in range(k)), 1) )})\n", " return nCk.dynamic[(n,k)]\n", "nCk.dynamic = {}\n", "\n", "# Simpler version...\n", "def nCr(n,r):\n", " f = math.factorial\n", " return f(n) / f(r) / f(n-r)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 216 }, { "cell_type": "code", "collapsed": false, "input": [ "from time import time\n", "t0 = time()\n", "nCk(100,50)\n", "print str(time() - t0)\n", "t0 = time()\n", "nCr(100,50)\n", "print str(time() - t0)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0.000294923782349\n", "0.000556945800781\n" ] } ], "prompt_number": 215 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Bezier from Definition" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def B(n,j, t):\n", " \"\"\" Bernstein Basis Function of degree n \"\"\"\n", " assert(0 <= j <= n)\n", " return nCk(n,j) * t**j * (1. - t)**(n-j)\n", "\n", "def Bezier(P, t):\n", " \"\"\" Apply Bezier definition to produce a point on the Bezier curve with control points P \"\"\"\n", " n = len(P)-1\n", " return [sum([B(n, j, t) * P[j][i] for j in xrange(n+1)]) for i in xrange(len(P[0]))]\n", " \n", "def PlotBezier(P,nPoints=50,style='o-'):\n", " \"\"\" Plot a Bezier \"\"\"\n", " points = map(lambda t : Bezier(P,t), linspace(0,1,nPoints))\n", " x = [p[0] for p in points]\n", " y = [p[1] for p in points]\n", " plot(x,y, style)\n", " plot([p[0] for p in P], [p[1] for p in P],'ro--')\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 266 }, { "cell_type": "code", "collapsed": false, "input": [ "title(\"Bezier Curves of Degree 2\")\n", "PlotBezier([(0,0),(0.4,0.8),(1,1)],nPoints=100,style='-')\n", "PlotBezier([(0,0),(0.4,-0.8),(1,1)],nPoints=100,style='-')\n", "legend()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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ETwhRPQUFQFzc20T+5t8ffii5u/3TT7wX36wZ78WbmUFWszYePwZu3+a3O3f4TSoFHBwA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\n8tDFtAu6mnZFV7OucDJxQr2a9YQOV3k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"text": [ "" ] } ], "prompt_number": 267 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Fractal Beziers" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$L P = Q$ and $M P = R$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def L(n):\n", " try:\n", " return L.dynamic[n]\n", " except KeyError:\n", " result = asarray([asarray([ nCk(j,k) / (2.**j) for k in xrange(n+1)]) for j in xrange(n+1)])\n", " L.dynamic.update({n : result})\n", " return result\n", "L.dynamic = {} \n", "\n", "def M(n):\n", " try:\n", " return M.dynamic[n]\n", " except KeyError:\n", " result = asarray([asarray([ nCk(n-j,n-k) / (2.**(n-j)) for k in xrange(n+1)]) for j in xrange(n+1)])\n", " M.dynamic.update({n : result})\n", " return result \n", "M.dynamic = {}\n", "\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 296 }, { "cell_type": "code", "collapsed": false, "input": [ "points = [(0,0), (0.5,1), (1,0)]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 298 }, { "cell_type": "code", "collapsed": false, "input": [ "def Casteljau(P, nPoints):\n", " while len(P) < nPoints:\n", " Q = dot(L(len(P)-1), P)\n", " R = dot(M(len(P)-1), P)\n", " P = list(Q) + list(R)[1:]\n", " return P\n", " " ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 329 }, { "cell_type": "code", "collapsed": false, "input": [ "points = [(0,0), (0.5,1), (1,0)]\n", "c = Casteljau(points,50)\n", "PlotBezier(points)\n", "plot([p[0] for p in c], [p[1] for p in c], 'go-')" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 332, "text": [ "[]" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 332 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, it sort of works but not giving the same curve as the vanilla method" ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }