Created
May 31, 2013 23:15
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Use least squares to find the cubic bezier equation of a hand drawn arc.
This ipython notebook uses sympy to derive the equations needed to setup the least squares fitting problem.
It then creates some noisy data on an arc and shows that the best fitting process works quite well.
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| { | |
| "metadata": { | |
| "name": "BezierLeastSquaresFit" | |
| }, | |
| "nbformat": 3, | |
| "nbformat_minor": 0, | |
| "worksheets": [ | |
| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h1>Use a method of least squares to find a 'best bezier fit' for a hand drawn arc.</h1>\n", | |
| "<h3>This notebook will investigate the cubic bezier function along with how to perform a least squares best fit to find the optimal control points for a cubic bezier curve that best matches a hand drawn curve.</h3>\n", | |
| "For more background information on bezier curves see : http://en.wikipedia.org/wiki/B%C3%A9zier_curve\n", | |
| "\n", | |
| "For an excellent tutorial on least squared fitting see : http://www.embedded.com/electrical-engineer-community/industry-blog/4027019/1/Why-all-the-math-\n", | |
| "\n", | |
| "Author : Luke Whittlesey \n", | |
| "\n", | |
| "email : myfirstname.mylastname@gmail.com" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>First setup this ipython notebook for symbolics using sympy and inline plotting using matplotlib.</h3>" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# we need some ipython magic commands to \n", | |
| "# 1. allow sympy to pretty print math equations inline\n", | |
| "# 2. use pylab for inline graphing\n", | |
| "%pylab inline\n", | |
| "%load_ext sympy.interactive.ipythonprinting" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "\n", | |
| "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n", | |
| "For more information, type 'help(pylab)'.\n" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 1 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>To make sure we understand what a cubic bezier curve is, let's create one numerically and then plot it.</h3>" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "def numerical_cubic_bezier_fn(control_pts, t):\n", | |
| " \"\"\" \n", | |
| " Numerically calculate an x,y point for the bezier curve at position t. \n", | |
| "\n", | |
| " Parameters: control_pts -- list of 4 (x,y) control points\n", | |
| " t -- 'time' value should be between 0 and 1\n", | |
| "\n", | |
| " Return: (xpt, ypt) -- tuple of x,y data on the curve\n", | |
| " \"\"\"\n", | |
| " check = ((0 <= t) and (t <= 1))\n", | |
| " assert check\n", | |
| " # define the actual cubic bezier equation here\n", | |
| " fn = lambda c,t : c[0]*(1 - t)**3 + c[1]*3*t*(1 - t)**2 + c[2]*3*t**2*(1 - t) + c[3]*t**3\n", | |
| " xs = [x for x,y in control_pts]\n", | |
| " ys = [y for x,y in control_pts]\n", | |
| " # now calculate the x,y position from the bezier equation\n", | |
| " xpt = fn(xs, t)\n", | |
| " ypt = fn(ys, t)\n", | |
| " return xpt, ypt\n", | |
| "\n", | |
| "# just to make sure that I understand, lets create a list of numbers for a specific bezier curve\n", | |
| "# arbitrarily define my 4 control points here\n", | |
| "p0 = (0.7,0.2); p1 = (2.3,1.0); p2 = (7.2,0.8); p3 = (10.0,0.0)\n", | |
| "# create a list of 1000 x,y points that traces over the length of this curve\n", | |
| "points = [numerical_cubic_bezier_fn(control_pts=[p0,p1,p2,p3], t=t) for t in arange(0.0,1.0,1./1000)]\n", | |
| "# separate the x's and y's so matplotlib can plot them\n", | |
| "xs = [x for x,y in points]\n", | |
| "ys = [y for x,y in points]\n", | |
| "\n", | |
| "# plot the resulting curve and the control points\n", | |
| "plot(xs, ys, 'b-')\n", | |
| "hold(True)\n", | |
| "for pt in [p0,p1,p2,p3]:\n", | |
| " plot( pt[0], pt[1], 'ro')\n", | |
| "hold(False)\n", | |
| "t=title('example cubic bezier curve')\n", | |
| "#legend(['curve','control pts'])" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "png": 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| |
| } | |
| ], | |
| "prompt_number": 2 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h2>\n", | |
| "Using sympy symbols we will now go through some mathy equations for how to setup the least squares problem of finding the best cubic bezier fit for our hand drawn data points.\n", | |
| "</h2> \n", | |
| "\n", | |
| "<h3>\n", | |
| "First redefine the cubic bezier equation using sympy.\n", | |
| "</h3>\n", | |
| "t must be defined in the range 0 to 1\n", | |
| "\\begin{aligned}\n", | |
| "t \\in \\begin{bmatrix}\n", | |
| "0 & 1\n", | |
| "\\end{bmatrix}\n", | |
| "\\end{aligned}" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from sympy.printing.mathml import mathml\n", | |
| "from sympy import Symbol, Matrix, summation, diff, expand, collect, preview, Sum, latex, Eq, factor\n", | |
| "\n", | |
| "t = Symbol('t_i')\n", | |
| "c0, c1, c2, c3 = [Symbol(name) for name in ['c0','c1','c2','c3']]\n", | |
| "\n", | |
| "# the equation for a cubic bezier path b(t) over t in [0,1]\n", | |
| "# Note: the c control points are (x,y) tuples, but we're going to compute\n", | |
| "# the resulting x and y values independantly. x and y are independant of\n", | |
| "# each other, but both are dependant on t which goes from 0 to 1\n", | |
| "symbolic_cubic_bezier_fn = c0*(1 - t)**3 + c1*3*t*(1 - t)**2 + c2*3*t**2*(1 - t) + c3*t**3\n", | |
| "print 'The cubic bezier function B(t) is:'\n", | |
| "B = Symbol('B(t_i)')\n", | |
| "Eq(B, symbolic_cubic_bezier_fn)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "The cubic bezier function B(t) is:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$B(t_{i)} = c_{0} \\left(- t_{i} + 1\\right)^{3} + 3 c_{1} t_{i} \\left(- t_{i} + 1\\right)^{2} + 3 c_{2} t_{i}^{2} \\left(- t_{i} + 1\\right) + c_{3} t_{i}^{3}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAewAAAAeCAYAAADuBy3GAAAABHNCSVQICAgIfAhkiAAACnxJREFU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\nD+3UrIIAXrrRAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 3, | |
| "text": [ | |
| " 3 2 2 3\n", | |
| "B(t\u1d62\u208e = c\u2080\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2081\u22c5t\u1d62\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2082\u22c5t\u1d62 \u22c5(-t\u1d62 + 1) + c\u2083\u22c5t\u1d62 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 3 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>\n", | |
| "Now we define the error for each data point as the difference between what the best fit bezier function gives us and what the actual data point is.\n", | |
| "</h3>\n", | |
| "\\begin{aligned}\n", | |
| "e_i = B(t_i) - d_i\n", | |
| "\\end{aligned}" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "d_i = Symbol('d_i')\n", | |
| "# the error between what the 'best fit' bezier curve produces and the hand drawn point is defined as\n", | |
| "error = symbolic_cubic_bezier_fn - d_i\n", | |
| "\n", | |
| "e_i = Symbol('e_i')\n", | |
| "# where symbolic_cubic_bezier_fn is bezier(t, control_points) and data_i is the i'th hand drawn point\n", | |
| "Eq(e_i, error)\n" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "latex": [ | |
| "$$e_{i} = c_{0} \\left(- t_{i} + 1\\right)^{3} + 3 c_{1} t_{i} \\left(- t_{i} + 1\\right)^{2} + 3 c_{2} t_{i}^{2} \\left(- t_{i} + 1\\right) + c_{3} t_{i}^{3} - d_{i}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAfQAAAAcCAYAAACXtk3RAAAABHNCSVQICAgIfAhkiAAACeBJREFU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\nDMMwDMMwDMMwDMMwjEb5Pz7bjRWUJr1UAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 4, | |
| "text": [ | |
| " 3 2 2 3 \n", | |
| "e\u1d62 = c\u2080\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2081\u22c5t\u1d62\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2082\u22c5t\u1d62 \u22c5(-t\u1d62 + 1) + c\u2083\u22c5t\u1d62 - d\u1d62" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 4 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>\n", | |
| "The penalty function will be the sum of all of the squared errors. I.e. least squares.\n", | |
| "</h3>\n", | |
| "\\begin{aligned}\n", | |
| "M = \\sum_{i=0}^{n} e_i^2\n", | |
| "\\end{aligned}\n", | |
| "This penalty function, M, is what we will want to minimize to call the solution a best fit." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# the least squares penalty function is the sum of all of the squared errors\n", | |
| "penalty_fn = error**2\n", | |
| "\n", | |
| "print 'The penalty function is :'\n", | |
| "i = Symbol('i')\n", | |
| "n = Symbol('n')\n", | |
| "M = Symbol('M')\n", | |
| "Eq(M, Sum( penalty_fn, (i,0,n) ))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "The penalty function is :\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$M = \\sum_{i=0}^{n} \\left(c_{0} \\left(- t_{i} + 1\\right)^{3} + 3 c_{1} t_{i} \\left(- t_{i} + 1\\right)^{2} + 3 c_{2} t_{i}^{2} \\left(- t_{i} + 1\\right) + c_{3} t_{i}^{3} - d_{i}\\right)^{2}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 5, | |
| "text": [ | |
| " n \n", | |
| " ____ \n", | |
| " \u2572 \n", | |
| " \u2572 \n", | |
| " \u2572 \u239b 3 2 2 3 \n", | |
| "M = \u2571 \u239dc\u2080\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2081\u22c5t\u1d62\u22c5(-t\u1d62 + 1) + 3\u22c5c\u2082\u22c5t\u1d62 \u22c5(-t\u1d62 + 1) + c\u2083\u22c5t\u1d62 - \n", | |
| " \u2571 \n", | |
| " \u2571 \n", | |
| " \u203e\u203e\u203e\u203e \n", | |
| " i = 0 \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 2\n", | |
| " \u239e \n", | |
| "d\u1d62\u23a0 \n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 5 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>We want to minimize the error in the penalty function for each coef c0,c1,c2,c3\n", | |
| " so take partial derivatives and find where they equal zero to find the minimum (or maximum). </h3>\n", | |
| "\\begin{aligned}\n", | |
| "\\frac{\\partial M}{\\partial c_0} = 0,\n", | |
| "\\frac{\\partial M}{\\partial c_1} = 0, \n", | |
| "\\frac{\\partial M}{\\partial c_2} = 0, \n", | |
| "\\frac{\\partial M}{\\partial c_3} = 0\n", | |
| "\\end{aligned}\n", | |
| "We actually know that this can only be a minimum since the maximum error is infinite and therefore the derivative won't have a zero crossing for the maximum." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "df_dc0 = collect( expand(penalty_fn.diff(c0), deep=True), [c0,c1,c2,c3])\n", | |
| "print 'Partial derivative wrt c0 is:'\n", | |
| "Eq(0, Sum(df_dc0, (i,0,n)) )" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "Partial derivative wrt c0 is:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$0 = \\sum_{i=0}^{n} \\left(c_{0} \\left(2 t_{i}^{6} - 12 t_{i}^{5} + 30 t_{i}^{4} - 40 t_{i}^{3} + 30 t_{i}^{2} - 12 t_{i} + 2\\right) + c_{1} \\left(- 6 t_{i}^{6} + 30 t_{i}^{5} - 60 t_{i}^{4} + 60 t_{i}^{3} - 30 t_{i}^{2} + 6 t_{i}\\right) + c_{2} \\left(6 t_{i}^{6} - 24 t_{i}^{5} + 36 t_{i}^{4} - 24 t_{i}^{3} + 6 t_{i}^{2}\\right) + c_{3} \\left(- 2 t_{i}^{6} + 6 t_{i}^{5} - 6 t_{i}^{4} + 2 t_{i}^{3}\\right) + 2 d_{i} t_{i}^{3} - 6 d_{i} t_{i}^{2} + 6 d_{i} t_{i} - 2 d_{i}\\right)$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 6, | |
| "text": [ | |
| " n \n", | |
| " ___ \n", | |
| " \u2572 \n", | |
| " \u2572 \u239b \u239b 6 5 4 3 2 \u239e \u239b \n", | |
| "0 = \u2571 \u239dc\u2080\u22c5\u239d2\u22c5t\u1d62 - 12\u22c5t\u1d62 + 30\u22c5t\u1d62 - 40\u22c5t\u1d62 + 30\u22c5t\u1d62 - 12\u22c5t\u1d62 + 2\u23a0 + c\u2081\u22c5\u239d- \n", | |
| " \u2571 \n", | |
| " \u203e\u203e\u203e \n", | |
| " i = 0 \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 6 5 4 3 2 \u239e \u239b 6 5 \n", | |
| "6\u22c5t\u1d62 + 30\u22c5t\u1d62 - 60\u22c5t\u1d62 + 60\u22c5t\u1d62 - 30\u22c5t\u1d62 + 6\u22c5t\u1d62\u23a0 + c\u2082\u22c5\u239d6\u22c5t\u1d62 - 24\u22c5t\u1d62 + 36\u22c5t\u1d62\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "4 3 2\u239e \u239b 6 5 4 3\u239e 3 \n", | |
| " - 24\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 + c\u2083\u22c5\u239d- 2\u22c5t\u1d62 + 6\u22c5t\u1d62 - 6\u22c5t\u1d62 + 2\u22c5t\u1d62 \u23a0 + 2\u22c5d\u1d62\u22c5t\u1d62 - 6\u22c5d\u1d62\u22c5t\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 2 \u239e\n", | |
| "\u1d62 + 6\u22c5d\u1d62\u22c5t\u1d62 - 2\u22c5d\u1d62\u23a0\n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 6 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "df_dc1 = collect( expand(penalty_fn.diff(c1), deep=True), [c0,c1,c2,c3])\n", | |
| "print 'Partial derivative wrt c1 is:'\n", | |
| "Eq(0, Sum(df_dc1, (i,0,n)))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "Partial derivative wrt c1 is:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$0 = \\sum_{i=0}^{n} \\left(c_{0} \\left(- 6 t_{i}^{6} + 30 t_{i}^{5} - 60 t_{i}^{4} + 60 t_{i}^{3} - 30 t_{i}^{2} + 6 t_{i}\\right) + c_{1} \\left(18 t_{i}^{6} - 72 t_{i}^{5} + 108 t_{i}^{4} - 72 t_{i}^{3} + 18 t_{i}^{2}\\right) + c_{2} \\left(- 18 t_{i}^{6} + 54 t_{i}^{5} - 54 t_{i}^{4} + 18 t_{i}^{3}\\right) + c_{3} \\left(6 t_{i}^{6} - 12 t_{i}^{5} + 6 t_{i}^{4}\\right) - 6 d_{i} t_{i}^{3} + 12 d_{i} t_{i}^{2} - 6 d_{i} t_{i}\\right)$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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ppyp5n6dsXC/DuoNNHtK5NRbndQ6/5XP75n2vjpwetfmVZxRy5J1Ye82F3tHlpEP7p3K+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\nIiIiIiIiIiIiIiIy4v4fRBDBpxTopXUAAAAASUVORK5CYII=\n", | |
| "prompt_number": 7, | |
| "text": [ | |
| " n \n", | |
| " ___ \n", | |
| " \u2572 \n", | |
| " \u2572 \u239b \u239b 6 5 4 3 2 \u239e \u239b \n", | |
| "0 = \u2571 \u239dc\u2080\u22c5\u239d- 6\u22c5t\u1d62 + 30\u22c5t\u1d62 - 60\u22c5t\u1d62 + 60\u22c5t\u1d62 - 30\u22c5t\u1d62 + 6\u22c5t\u1d62\u23a0 + c\u2081\u22c5\u239d18\u22c5t\u1d62\n", | |
| " \u2571 \n", | |
| " \u203e\u203e\u203e \n", | |
| " i = 0 \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "6 5 4 3 2\u239e \u239b 6 5 4 \n", | |
| " - 72\u22c5t\u1d62 + 108\u22c5t\u1d62 - 72\u22c5t\u1d62 + 18\u22c5t\u1d62 \u23a0 + c\u2082\u22c5\u239d- 18\u22c5t\u1d62 + 54\u22c5t\u1d62 - 54\u22c5t\u1d62 + 18\u22c5\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 3\u239e \u239b 6 5 4\u239e 3 2 \u239e\n", | |
| "t\u1d62 \u23a0 + c\u2083\u22c5\u239d6\u22c5t\u1d62 - 12\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 - 6\u22c5d\u1d62\u22c5t\u1d62 + 12\u22c5d\u1d62\u22c5t\u1d62 - 6\u22c5d\u1d62\u22c5t\u1d62\u23a0\n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 7 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "df_dc2 = collect( expand(penalty_fn.diff(c2), deep=True), [c0,c1,c2,c3])\n", | |
| "print 'Partial derivative wrt c2 is:'\n", | |
| "Eq(0, Sum(df_dc2, (i,0,n)) )" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "Partial derivative wrt c2 is:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$0 = \\sum_{i=0}^{n} \\left(c_{0} \\left(6 t_{i}^{6} - 24 t_{i}^{5} + 36 t_{i}^{4} - 24 t_{i}^{3} + 6 t_{i}^{2}\\right) + c_{1} \\left(- 18 t_{i}^{6} + 54 t_{i}^{5} - 54 t_{i}^{4} + 18 t_{i}^{3}\\right) + c_{2} \\left(18 t_{i}^{6} - 36 t_{i}^{5} + 18 t_{i}^{4}\\right) + c_{3} \\left(- 6 t_{i}^{6} + 6 t_{i}^{5}\\right) + 6 d_{i} t_{i}^{3} - 6 d_{i} t_{i}^{2}\\right)$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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LZafTOUZ9x/tYIH4jYv2VWIAeiw1otTXWSpi1IkfZBriK5A1wjwHHhpb1AN/Cntn7Qui9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\nAAAASUVORK5CYII=\n", | |
| "prompt_number": 8, | |
| "text": [ | |
| " n \n", | |
| " ___ \n", | |
| " \u2572 \n", | |
| " \u2572 \u239b \u239b 6 5 4 3 2\u239e \u239b 6 \n", | |
| "0 = \u2571 \u239dc\u2080\u22c5\u239d6\u22c5t\u1d62 - 24\u22c5t\u1d62 + 36\u22c5t\u1d62 - 24\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 + c\u2081\u22c5\u239d- 18\u22c5t\u1d62 + 54\u22c5t\n", | |
| " \u2571 \n", | |
| " \u203e\u203e\u203e \n", | |
| " i = 0 \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 5 4 3\u239e \u239b 6 5 4\u239e \u239b 6 5\u239e \n", | |
| "\u1d62 - 54\u22c5t\u1d62 + 18\u22c5t\u1d62 \u23a0 + c\u2082\u22c5\u239d18\u22c5t\u1d62 - 36\u22c5t\u1d62 + 18\u22c5t\u1d62 \u23a0 + c\u2083\u22c5\u239d- 6\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 +\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " 3 2\u239e\n", | |
| " 6\u22c5d\u1d62\u22c5t\u1d62 - 6\u22c5d\u1d62\u22c5t\u1d62 \u23a0\n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 8 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "df_dc3 = collect( expand(penalty_fn.diff(c3), deep=True), [c0,c1,c2,c3])\n", | |
| "print 'Partial derivative wrt c3 is:'\n", | |
| "Eq(0, Sum(df_dc3, (i,0,n)) )" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "Partial derivative wrt c3 is:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$0 = \\sum_{i=0}^{n} \\left(c_{0} \\left(- 2 t_{i}^{6} + 6 t_{i}^{5} - 6 t_{i}^{4} + 2 t_{i}^{3}\\right) + c_{1} \\left(6 t_{i}^{6} - 12 t_{i}^{5} + 6 t_{i}^{4}\\right) + c_{2} \\left(- 6 t_{i}^{6} + 6 t_{i}^{5}\\right) + 2 c_{3} t_{i}^{6} - 2 d_{i} t_{i}^{3}\\right)$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 9, | |
| "text": [ | |
| " n \n", | |
| " ___ \n", | |
| " \u2572 \n", | |
| " \u2572 \u239b \u239b 6 5 4 3\u239e \u239b 6 5 4\u239e \n", | |
| "0 = \u2571 \u239dc\u2080\u22c5\u239d- 2\u22c5t\u1d62 + 6\u22c5t\u1d62 - 6\u22c5t\u1d62 + 2\u22c5t\u1d62 \u23a0 + c\u2081\u22c5\u239d6\u22c5t\u1d62 - 12\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 \n", | |
| " \u2571 \n", | |
| " \u203e\u203e\u203e \n", | |
| " i = 0 \n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| " \u239b 6 5\u239e 6 3\u239e\n", | |
| "+ c\u2082\u22c5\u239d- 6\u22c5t\u1d62 + 6\u22c5t\u1d62 \u23a0 + 2\u22c5c\u2083\u22c5t\u1d62 - 2\u22c5d\u1d62\u22c5t\u1d62 \u23a0\n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 9 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Notice that all of the partial derivatives only have a single power for c0,c1,c2,c3. This means that these equations are linear with respect to the c's. To minimize the error we have to set each partial derivative to zero, to find the minima, and then solve 4 simultaneous equations. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>Now we should have 4 equations and 4 unknowns. We can use linear algebra to solve this for us, but first we need to reshuffle the equations to prepare them for matrix form.</h3>\n", | |
| "We will also factor to reduce some of the terms." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "def pull_arguments_by_var(eqn, variables):\n", | |
| " \"\"\"\n", | |
| " For a equation such as : a*2 + b*3*x + 5*x + 5\n", | |
| " this function will return a dictionary of the arguments that are multiplied by each variable.\n", | |
| " So if variables=[a,b], the dict returned would be:\n", | |
| " a:[2],\n", | |
| " b:[3*x],\n", | |
| " 'other':[5*x, 5]\n", | |
| "\n", | |
| " Parameters: eqn -- a sympy equation that has already been term collected\n", | |
| " variables -- a list of variables to search for arguments that are multiplied by each variable\n", | |
| "\n", | |
| " Returns: vardict -- a dictionary containing the arguments for each variable and 'other'\n", | |
| " \"\"\"\n", | |
| " # try to find what is multiplied by the variables in variables and other\n", | |
| " cdict = {}\n", | |
| " for var in variables:\n", | |
| " for arg in eqn.args:\n", | |
| " if var in arg.atoms(Symbol):\n", | |
| " cdict.setdefault(var, []).append(arg/var)\n", | |
| " #print '%s : %s' %(c, arg)\n", | |
| " for arg in eqn.args:\n", | |
| " # test for non-cx arguments\n", | |
| " if not any([(cc in arg) for cc in variables]):\n", | |
| " cdict.setdefault('other', []).append(arg)\n", | |
| " #print 'other : %s' %(arg)\n", | |
| " return cdict " | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 10 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# from the 1st partial derivative, reshuffle the equations so the c's are on the left and the others are on the right\n", | |
| "# also redistribute the sums\n", | |
| "cvars = [c0,c1,c2,c3]\n", | |
| "arg_dict = pull_arguments_by_var(eqn=df_dc0, variables=cvars)\n", | |
| "leftside = c0*Sum(factor(arg_dict[c0][0]), (i,0,n)) + \\\n", | |
| " c1*Sum(factor(arg_dict[c1][0]), (i,0,n)) + \\\n", | |
| " c2*Sum(factor(arg_dict[c2][0]), (i,0,n)) + \\\n", | |
| " c3*Sum(factor(arg_dict[c3][0]), (i,0,n))\n", | |
| "rightside = arg_dict['other'][0]\n", | |
| "for arg in arg_dict['other'][1:]:\n", | |
| " rightside = rightside + arg\n", | |
| "print '1st Equation:'\n", | |
| "Eq(leftside, -1*Sum(factor(rightside), (i,0,n)))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "1st Equation:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$c_{0} \\sum_{i=0}^{n} 2 \\left(t_{i} -1\\right)^{6} + c_{1} \\sum_{i=0}^{n} - 6 t_{i} \\left(t_{i} -1\\right)^{5} + c_{2} \\sum_{i=0}^{n} 6 t_{i}^{2} \\left(t_{i} -1\\right)^{4} + c_{3} \\sum_{i=0}^{n} - 2 t_{i}^{3} \\left(t_{i} -1\\right)^{3} = - \\sum_{i=0}^{n} 2 d_{i} \\left(t_{i} -1\\right)^{3}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 11, | |
| "text": [ | |
| " n n n \n", | |
| " ___ ___ ___ \n", | |
| " \u2572 \u2572 \u2572 \n", | |
| " \u2572 6 \u2572 5 \u2572 2 4 \n", | |
| "c\u2080\u22c5 \u2571 2\u22c5(t\u1d62 - 1) + c\u2081\u22c5 \u2571 -6\u22c5t\u1d62\u22c5(t\u1d62 - 1) + c\u2082\u22c5 \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\n", | |
| " \u2571 \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 i = 0 \n", | |
| "\n", | |
| " n n \n", | |
| " ___ ___ \n", | |
| " \u2572 \u2572 \n", | |
| " \u2572 3 3 \u2572 3\n", | |
| "\u2083\u22c5 \u2571 -2\u22c5t\u1d62 \u22c5(t\u1d62 - 1) = - \u2571 2\u22c5d\u1d62\u22c5(t\u1d62 - 1) \n", | |
| " \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 11 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# from the 2nd partial derivative, reshuffle the equations so the c's are on the left and the others are on the right\n", | |
| "# also redistribute the sums\n", | |
| "cvars = [c0,c1,c2,c3]\n", | |
| "arg_dict = pull_arguments_by_var(eqn=df_dc1, variables=cvars)\n", | |
| "leftside = c0*Sum(factor(arg_dict[c0][0]), (i,0,n)) + \\\n", | |
| " c1*Sum(factor(arg_dict[c1][0]), (i,0,n)) + \\\n", | |
| " c2*Sum(factor(arg_dict[c2][0]), (i,0,n)) + \\\n", | |
| " c3*Sum(factor(arg_dict[c3][0]), (i,0,n))\n", | |
| "rightside = arg_dict['other'][0]\n", | |
| "for arg in arg_dict['other'][1:]:\n", | |
| " rightside = rightside + arg\n", | |
| "print '2nd Equation:'\n", | |
| "Eq(leftside, -1*Sum(factor(rightside), (i,0,n)))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "2nd Equation:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$c_{0} \\sum_{i=0}^{n} - 6 t_{i} \\left(t_{i} -1\\right)^{5} + c_{1} \\sum_{i=0}^{n} 18 t_{i}^{2} \\left(t_{i} -1\\right)^{4} + c_{2} \\sum_{i=0}^{n} - 18 t_{i}^{3} \\left(t_{i} -1\\right)^{3} + c_{3} \\sum_{i=0}^{n} 6 t_{i}^{4} \\left(t_{i} -1\\right)^{2} = - \\sum_{i=0}^{n} - 6 d_{i} t_{i} \\left(t_{i} -1\\right)^{2}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 12, | |
| "text": [ | |
| " n n n \n", | |
| " ___ ___ ___ \n", | |
| " \u2572 \u2572 \u2572 \n", | |
| " \u2572 5 \u2572 2 4 \u2572 3 \n", | |
| "c\u2080\u22c5 \u2571 -6\u22c5t\u1d62\u22c5(t\u1d62 - 1) + c\u2081\u22c5 \u2571 18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\u2082\u22c5 \u2571 -18\u22c5t\u1d62 \u22c5(t\u1d62 - \n", | |
| " \u2571 \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 i = 0 \n", | |
| "\n", | |
| " n n \n", | |
| " ___ ___ \n", | |
| " \u2572 \u2572 \n", | |
| " 3 \u2572 4 2 \u2572 2\n", | |
| "1) + c\u2083\u22c5 \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) = - \u2571 -6\u22c5d\u1d62\u22c5t\u1d62\u22c5(t\u1d62 - 1) \n", | |
| " \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 12 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# from the 3rd partial derivative, reshuffle the equations so the c's are on the left and the others are on the right\n", | |
| "# also redistribute the sums\n", | |
| "cvars = [c0,c1,c2,c3]\n", | |
| "arg_dict = pull_arguments_by_var(eqn=df_dc2, variables=cvars)\n", | |
| "leftside = c0*Sum(factor(arg_dict[c0][0]), (i,0,n)) + \\\n", | |
| " c1*Sum(factor(arg_dict[c1][0]), (i,0,n)) + \\\n", | |
| " c2*Sum(factor(arg_dict[c2][0]), (i,0,n)) + \\\n", | |
| " c3*Sum(factor(arg_dict[c3][0]), (i,0,n))\n", | |
| "rightside = arg_dict['other'][0]\n", | |
| "for arg in arg_dict['other'][1:]:\n", | |
| " rightside = rightside + arg\n", | |
| "print '3rd Equation:'\n", | |
| "Eq(leftside, -1*Sum(factor(rightside), (i,0,n)))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "3rd Equation:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$c_{0} \\sum_{i=0}^{n} 6 t_{i}^{2} \\left(t_{i} -1\\right)^{4} + c_{1} \\sum_{i=0}^{n} - 18 t_{i}^{3} \\left(t_{i} -1\\right)^{3} + c_{2} \\sum_{i=0}^{n} 18 t_{i}^{4} \\left(t_{i} -1\\right)^{2} + c_{3} \\sum_{i=0}^{n} - 6 t_{i}^{5} \\left(t_{i} -1\\right) = - \\sum_{i=0}^{n} 6 d_{i} t_{i}^{2} \\left(t_{i} -1\\right)$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": 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| |
| "prompt_number": 13, | |
| "text": [ | |
| " n n n \n", | |
| " ___ ___ ___ \n", | |
| " \u2572 \u2572 \u2572 \n", | |
| " \u2572 2 4 \u2572 3 3 \u2572 4 \n", | |
| "c\u2080\u22c5 \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\u2081\u22c5 \u2571 -18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\u2082\u22c5 \u2571 18\u22c5t\u1d62 \u22c5(t\u1d62 - \n", | |
| " \u2571 \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 i = 0 \n", | |
| "\n", | |
| " n n \n", | |
| " ___ ___ \n", | |
| " \u2572 \u2572 \n", | |
| " 2 \u2572 5 \u2572 2 \n", | |
| "1) + c\u2083\u22c5 \u2571 -6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) = - \u2571 6\u22c5d\u1d62\u22c5t\u1d62 \u22c5(t\u1d62 - 1)\n", | |
| " \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 13 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# from the 4th partial derivative, reshuffle the equations so the c's are on the left and the others are on the right\n", | |
| "# also redistribute the sums\n", | |
| "cvars = [c0,c1,c2,c3]\n", | |
| "arg_dict = pull_arguments_by_var(eqn=df_dc3, variables=cvars)\n", | |
| "leftside = c0*Sum(factor(arg_dict[c0][0]), (i,0,n)) + \\\n", | |
| " c1*Sum(factor(arg_dict[c1][0]), (i,0,n)) + \\\n", | |
| " c2*Sum(factor(arg_dict[c2][0]), (i,0,n)) + \\\n", | |
| " c3*Sum(factor(arg_dict[c3][0]), (i,0,n))\n", | |
| "rightside = arg_dict['other'][0]\n", | |
| "for arg in arg_dict['other'][1:]:\n", | |
| " rightside = rightside + arg\n", | |
| "print '4th Equation:'\n", | |
| "Eq(leftside, -1*Sum(factor(rightside), (i,0,n)))" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "4th Equation:\n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$c_{0} \\sum_{i=0}^{n} - 2 t_{i}^{3} \\left(t_{i} -1\\right)^{3} + c_{1} \\sum_{i=0}^{n} 6 t_{i}^{4} \\left(t_{i} -1\\right)^{2} + c_{2} \\sum_{i=0}^{n} - 6 t_{i}^{5} \\left(t_{i} -1\\right) + c_{3} \\sum_{i=0}^{n} 2 t_{i}^{6} = - \\sum_{i=0}^{n} - 2 d_{i} t_{i}^{3}$$" | |
| ], | |
| "output_type": "pyout", | |
| "png": "iVBORw0KGgoAAAANSUhEUgAAAqgAAAA4CAYAAAAxQQo2AAAABHNCSVQICAgIfAhkiAAAEMRJREFU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\nCCGEEEIIIYQQQkTk/1A0AHb0SzoLAAAAAElFTkSuQmCC\n", | |
| "prompt_number": 14, | |
| "text": [ | |
| " n n n \n", | |
| " ___ ___ ___ \n", | |
| " \u2572 \u2572 \u2572 \n", | |
| " \u2572 3 3 \u2572 4 2 \u2572 5 \n", | |
| "c\u2080\u22c5 \u2571 -2\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\u2081\u22c5 \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) + c\u2082\u22c5 \u2571 -6\u22c5t\u1d62 \u22c5(t\u1d62 - 1\n", | |
| " \u2571 \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 i = 0 \n", | |
| "\n", | |
| " n n \n", | |
| " ___ ___ \n", | |
| " \u2572 \u2572 \n", | |
| " \u2572 6 \u2572 3\n", | |
| ") + c\u2083\u22c5 \u2571 2\u22c5t\u1d62 = - \u2571 -2\u22c5d\u1d62\u22c5t\u1d62 \n", | |
| " \u2571 \u2571 \n", | |
| " \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| " i = 0 i = 0 " | |
| ] | |
| } | |
| ], | |
| "prompt_number": 14 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>From the 4 equations above we can start to see the structure of the matrix form.</h3>\n", | |
| "Set the problem up as the familiar multiple linear equations, multiple unknowns pattern.\n", | |
| "\\begin{aligned}\n", | |
| "\\mathbf{A x = b}\n", | |
| "\\end{aligned}\n", | |
| "Where:\n", | |
| "\n", | |
| "$\\mathbf{A}$ is a 4x4 matrix of the summations listed on the left side of the equations\n", | |
| "\n", | |
| "$\\mathbf{x}$ is a 4x1 matrix containing the unknowns c0,c1,c2,c3\n", | |
| "\n", | |
| "$\\mathbf{b}$ is a 4x1 matrix of the summations listed on the right side of the equations\n", | |
| "\n", | |
| "To solve is simply : $\\mathbf{x = A^{-1} b}$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# build the actual matrices this time in a loop\n", | |
| "cvars = [c0,c1,c2,c3]\n", | |
| "a_vals = []\n", | |
| "x_vals = cvars\n", | |
| "b_vals = []\n", | |
| "for eqn in [df_dc0, df_dc1, df_dc2, df_dc3]:\n", | |
| " arg_dict = pull_arguments_by_var(eqn=eqn, variables=cvars)\n", | |
| " for var in cvars:\n", | |
| " a_vals.append( Sum(factor(arg_dict[var][0]), (i,0,n)) )\n", | |
| " rightside = arg_dict['other'][0]\n", | |
| " for arg in arg_dict['other'][1:]:\n", | |
| " rightside = rightside + arg\n", | |
| " b_vals.append( -1*Sum(factor(rightside), (i,0,n)) )\n", | |
| " \n", | |
| " \n", | |
| "A = Matrix(4,4,a_vals)\n", | |
| "x = Matrix(4,1,x_vals)\n", | |
| "b = Matrix(4,1,b_vals)" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 15 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "print 'A = '\n", | |
| "A" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "A = \n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$\\left[\\begin{smallmatrix}\\sum_{i=0}^{n} 2 \\left(t_{i} -1\\right)^{6} & \\sum_{i=0}^{n} - 6 t_{i} \\left(t_{i} -1\\right)^{5} & \\sum_{i=0}^{n} 6 t_{i}^{2} \\left(t_{i} -1\\right)^{4} & \\sum_{i=0}^{n} - 2 t_{i}^{3} \\left(t_{i} -1\\right)^{3}\\\\\\sum_{i=0}^{n} - 6 t_{i} \\left(t_{i} -1\\right)^{5} & \\sum_{i=0}^{n} 18 t_{i}^{2} \\left(t_{i} -1\\right)^{4} & \\sum_{i=0}^{n} - 18 t_{i}^{3} \\left(t_{i} -1\\right)^{3} & \\sum_{i=0}^{n} 6 t_{i}^{4} \\left(t_{i} -1\\right)^{2}\\\\\\sum_{i=0}^{n} 6 t_{i}^{2} \\left(t_{i} -1\\right)^{4} & \\sum_{i=0}^{n} - 18 t_{i}^{3} \\left(t_{i} -1\\right)^{3} & \\sum_{i=0}^{n} 18 t_{i}^{4} \\left(t_{i} -1\\right)^{2} & \\sum_{i=0}^{n} - 6 t_{i}^{5} \\left(t_{i} -1\\right)\\\\\\sum_{i=0}^{n} - 2 t_{i}^{3} \\left(t_{i} -1\\right)^{3} & \\sum_{i=0}^{n} 6 t_{i}^{4} \\left(t_{i} -1\\right)^{2} & \\sum_{i=0}^{n} - 6 t_{i}^{5} \\left(t_{i} -1\\right) & \\sum_{i=0}^{n} 2 t_{i}^{6}\\end{smallmatrix}\\right]$$" | |
| ], | |
| "output_type": "pyout", | |
| "prompt_number": 16, | |
| "text": [ | |
| "\u23a1 n n n n\n", | |
| "\u23a2 ___ ___ ___ __\n", | |
| "\u23a2 \u2572 \u2572 \u2572 \u2572 \n", | |
| "\u23a2 \u2572 6 \u2572 5 \u2572 2 4 \u2572\n", | |
| "\u23a2 \u2571 2\u22c5(t\u1d62 - 1) \u2571 -6\u22c5t\u1d62\u22c5(t\u1d62 - 1) \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571\n", | |
| "\u23a2 \u2571 \u2571 \u2571 \u2571 \n", | |
| "\u23a2 \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\n", | |
| "\u23a2 i = 0 i = 0 i = 0 i =\n", | |
| "\u23a2 \n", | |
| "\u23a2 n n n n\n", | |
| "\u23a2 ___ ___ ___ __\n", | |
| "\u23a2 \u2572 \u2572 \u2572 \u2572 \n", | |
| "\u23a2 \u2572 5 \u2572 2 4 \u2572 3 3 \u2572\n", | |
| "\u23a2 \u2571 -6\u22c5t\u1d62\u22c5(t\u1d62 - 1) \u2571 18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571 -18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571\n", | |
| "\u23a2 \u2571 \u2571 \u2571 \u2571 \n", | |
| "\u23a2 \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\n", | |
| "\u23a2i = 0 i = 0 i = 0 i =\n", | |
| "\u23a2 \n", | |
| "\u23a2 n n n n\n", | |
| "\u23a2 ___ ___ ___ __\n", | |
| "\u23a2 \u2572 \u2572 \u2572 \u2572 \n", | |
| "\u23a2 \u2572 2 4 \u2572 3 3 \u2572 4 2 \u2572\n", | |
| "\u23a2 \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571 -18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571 18\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571\n", | |
| "\u23a2 \u2571 \u2571 \u2571 \u2571 \n", | |
| "\u23a2 \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\n", | |
| "\u23a2i = 0 i = 0 i = 0 i =\n", | |
| "\u23a2 \n", | |
| "\u23a2 n n n \n", | |
| "\u23a2 ___ ___ ___ \n", | |
| "\u23a2 \u2572 \u2572 \u2572 \n", | |
| "\u23a2 \u2572 3 3 \u2572 4 2 \u2572 5 \n", | |
| "\u23a2 \u2571 -2\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u2571 -6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \n", | |
| "\u23a2 \u2571 \u2571 \u2571 \n", | |
| "\u23a2 \u203e\u203e\u203e \u203e\u203e\u203e \u203e\u203e\u203e \n", | |
| "\u23a3i = 0 i = 0 i = 0 \n", | |
| "\n", | |
| " \u23a4\n", | |
| "_ \u23a5\n", | |
| " \u23a5\n", | |
| " 3 3\u23a5\n", | |
| " -2\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u23a5\n", | |
| " \u23a5\n", | |
| "\u203e \u23a5\n", | |
| " 0 \u23a5\n", | |
| " \u23a5\n", | |
| " \u23a5\n", | |
| "_ \u23a5\n", | |
| " \u23a5\n", | |
| " 4 2 \u23a5\n", | |
| " 6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u23a5\n", | |
| " \u23a5\n", | |
| "\u203e \u23a5\n", | |
| " 0 \u23a5\n", | |
| " \u23a5\n", | |
| " \u23a5\n", | |
| "_ \u23a5\n", | |
| " \u23a5\n", | |
| " 5 \u23a5\n", | |
| " -6\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u23a5\n", | |
| " \u23a5\n", | |
| "\u203e \u23a5\n", | |
| " 0 \u23a5\n", | |
| " \u23a5\n", | |
| " n \u23a5\n", | |
| " ___ \u23a5\n", | |
| " \u2572 \u23a5\n", | |
| " \u2572 6 \u23a5\n", | |
| " \u2571 2\u22c5t\u1d62 \u23a5\n", | |
| " \u2571 \u23a5\n", | |
| " \u203e\u203e\u203e \u23a5\n", | |
| " i = 0 \u23a6" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 16 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "print 'x = '\n", | |
| "x" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "x = \n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$\\left[\\begin{smallmatrix}c_{0}\\\\c_{1}\\\\c_{2}\\\\c_{3}\\end{smallmatrix}\\right]$$" | |
| ], | |
| "output_type": "pyout", | |
| "prompt_number": 17, | |
| "text": [ | |
| "\u23a1c\u2080\u23a4\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2c\u2081\u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2c\u2082\u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a3c\u2083\u23a6" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 17 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "print 'b = '\n", | |
| "b" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "b = \n" | |
| ] | |
| }, | |
| { | |
| "latex": [ | |
| "$$\\left[\\begin{smallmatrix}- \\sum_{i=0}^{n} 2 d_{i} \\left(t_{i} -1\\right)^{3}\\\\- \\sum_{i=0}^{n} - 6 d_{i} t_{i} \\left(t_{i} -1\\right)^{2}\\\\- \\sum_{i=0}^{n} 6 d_{i} t_{i}^{2} \\left(t_{i} -1\\right)\\\\- \\sum_{i=0}^{n} - 2 d_{i} t_{i}^{3}\\end{smallmatrix}\\right]$$" | |
| ], | |
| "output_type": "pyout", | |
| "prompt_number": 18, | |
| "text": [ | |
| "\u23a1 n \u23a4\n", | |
| "\u23a2 ___ \u23a5\n", | |
| "\u23a2 \u2572 \u23a5\n", | |
| "\u23a2 \u2572 3 \u23a5\n", | |
| "\u23a2 - \u2571 2\u22c5d\u1d62\u22c5(t\u1d62 - 1) \u23a5\n", | |
| "\u23a2 \u2571 \u23a5\n", | |
| "\u23a2 \u203e\u203e\u203e \u23a5\n", | |
| "\u23a2 i = 0 \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 n \u23a5\n", | |
| "\u23a2 ___ \u23a5\n", | |
| "\u23a2 \u2572 \u23a5\n", | |
| "\u23a2 \u2572 2\u23a5\n", | |
| "\u23a2- \u2571 -6\u22c5d\u1d62\u22c5t\u1d62\u22c5(t\u1d62 - 1) \u23a5\n", | |
| "\u23a2 \u2571 \u23a5\n", | |
| "\u23a2 \u203e\u203e\u203e \u23a5\n", | |
| "\u23a2 i = 0 \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 n \u23a5\n", | |
| "\u23a2 ___ \u23a5\n", | |
| "\u23a2 \u2572 \u23a5\n", | |
| "\u23a2 \u2572 2 \u23a5\n", | |
| "\u23a2- \u2571 6\u22c5d\u1d62\u22c5t\u1d62 \u22c5(t\u1d62 - 1) \u23a5\n", | |
| "\u23a2 \u2571 \u23a5\n", | |
| "\u23a2 \u203e\u203e\u203e \u23a5\n", | |
| "\u23a2 i = 0 \u23a5\n", | |
| "\u23a2 \u23a5\n", | |
| "\u23a2 n \u23a5\n", | |
| "\u23a2 ___ \u23a5\n", | |
| "\u23a2 \u2572 \u23a5\n", | |
| "\u23a2 \u2572 3 \u23a5\n", | |
| "\u23a2 - \u2571 -2\u22c5d\u1d62\u22c5t\u1d62 \u23a5\n", | |
| "\u23a2 \u2571 \u23a5\n", | |
| "\u23a2 \u203e\u203e\u203e \u23a5\n", | |
| "\u23a3 i = 0 \u23a6" | |
| ] | |
| } | |
| ], | |
| "prompt_number": 18 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>Now we're done with symbolic math.</h3>\n", | |
| "The A and b matrices still look formidable with those summations in there. Do realize however that the summations are computed from the data prior to solving for c0,c1,c2, and c3. As far as the c values are concerned, the summations involving d_i's and t_i's just come out to be numbers computed from your hand drawn data. This means that the matrix math is actually quite simple. You use your d_i and t_i values in the summations to place numbers into the 4x4 A matrix and the 4x1 b matrix.\n", | |
| "From here we can do our matrix math numerically." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h2>Now it's time to see if we can actually compute a bezier curve from hand drawn data.<h2>\n", | |
| " <h3>This is not a gui program to draw an arc with, so add some noise to our original curve to simulate a poorly drawn line.</h3>\n", | |
| "Remember that when we started this worksheet, somewhere near the top, we had computed a bezier curve numerically. We're going to do the same thing here, but add noise to the output curve data to simulate our hand drawing." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "from pylab import randn\n", | |
| "\n", | |
| "# define the control points of a bezier curve we wish to generate\n", | |
| "p0 = (0.7,0.2); p1 = (2.3,1.0); p2 = (7.2,0.8); p3 = (10.0,0.0)\n", | |
| "# create a list of 1000 x,y points that traces over the length of this curve\n", | |
| "points = [numerical_cubic_bezier_fn(control_pts=[p0,p1,p2,p3], t=t) for t in arange(0.0,1.0,1./1000)]\n", | |
| "# separate the x's and y's so matplotlib can plot them; also add some noise\n", | |
| "xs = [x+0.15*randn() for x,y in points]\n", | |
| "ys = [y+0.15*randn() for x,y in points]\n", | |
| "\n", | |
| "# plot the resulting 'hand drawn' curve\n", | |
| "plot(xs, ys, 'b-')\n", | |
| "t=title('noisy (hand drawn) curve')" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "png": 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gWhr9X0wM9bpt3VqUL2AL3K7RCMs7ISH2mv4ff9C+OZ6dByMZxtLGduCM3jZt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| |
| } | |
| ], | |
| "prompt_number": 19 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h3>Now it's time for the numerical code for all of those summations that we computed symbolically above.</h3>\n", | |
| "First we have to define what $t_i$ and $d_i$ are... \n", | |
| "\n", | |
| "We used $d_i$ to represent each of our drawing data points in sequence, but you might be confused on how we are going to use (x,y) points for each $d_i$. The answer is that we will compute only for x and then we will repeat the process only for y. This should give us the x portion of the control points and then the y portion of the control points. Note that x and y are orthogonal and so they are independant of each other, so we can get away with calculating them independantly.\n", | |
| "\n", | |
| "For $t_i$, just imagine that someone is drawing the curve starting at one position and ending at another position in one continuous line. For this case we will assume it was drawn from the left and was finished on the right. $t_i$ is simply the time at which each data point was drawn. Remember that $t_i$ needs to go from 0 to 1 for the bezier equation to work. So, we assume that it took 1 second to draw the entire curve and each data point is drawn at the same relative speed. Of course time actually has nothing to do with anything, it's just a mental model of tying a $t_i$ to a $d_i$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# we already have out d_i's in xs and ys from above\n", | |
| "# make a t_i list\n", | |
| "ts = arange(0.0, 1.0, 1.0/len(points))\n", | |
| "\n", | |
| "# make the summation functions for A (16 of them)\n", | |
| "A_fns = [None for i in range(16)]\n", | |
| "A_fns[0] = lambda time_list, data_list : sum([2*(t_i - 1)**6 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[1] = lambda time_list, data_list : sum([-6*t_i*(t_i - 1)**5 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[2] = lambda time_list, data_list : sum([6*t_i**2*(t_i - 1)**4 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[3] = lambda time_list, data_list : sum([-2*t_i**3*(t_i - 1)**3 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[4] = lambda time_list, data_list : sum([-6*t_i*(t_i - 1)**5 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[5] = lambda time_list, data_list : sum([18*t_i**2*(t_i - 1)**4 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[6] = lambda time_list, data_list : sum([-18*t_i**3*(t_i - 1)**3 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[7] = lambda time_list, data_list : sum([6*t_i**4*(t_i - 1)**2 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[8] = lambda time_list, data_list : sum([6*t_i**2*(t_i - 1)**4 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[9] = lambda time_list, data_list : sum([-18*t_i**3*(t_i - 1)**3 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[10] = lambda time_list, data_list : sum([18*t_i**4*(t_i - 1)**2 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[11] = lambda time_list, data_list : sum([-6*t_i**5*(t_i - 1) for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[12] = lambda time_list, data_list : sum([-2*t_i**3*(t_i - 1)**3 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[13] = lambda time_list, data_list : sum([6*t_i**4*(t_i - 1)**2 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[14] = lambda time_list, data_list : sum([-6*t_i**5*(t_i - 1) for t_i, d_i in zip(time_list, data_list)])\n", | |
| "A_fns[15] = lambda time_list, data_list : sum([2*t_i**6 for t_i, d_i in zip(time_list, data_list)])\n", | |
| " \n", | |
| "# make the summation functions for b (4 of them)\n", | |
| "b_fns = [None for i in range(4)]\n", | |
| "b_fns[0] = lambda time_list, data_list : -1.0 * sum([2*d_i*(t_i - 1)**3 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "b_fns[1] = lambda time_list, data_list : -1.0 * sum([-6*d_i*t_i*(t_i - 1)**2 for t_i, d_i in zip(time_list, data_list)])\n", | |
| "b_fns[2] = lambda time_list, data_list : -1.0 * sum([6*d_i*t_i**2*(t_i - 1) for t_i, d_i in zip(time_list, data_list)])\n", | |
| "b_fns[3] = lambda time_list, data_list : -1.0 * sum([-2*d_i*t_i**3 for t_i, d_i in zip(time_list, data_list)])" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 20 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "From the generated 'hand drawn data' create a function to fill the A and b matrices using the summation functions we just defined and then solve for the unknowns. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "def solve_for_cs(time_series, data_series):\n", | |
| " \"\"\"\n", | |
| " Take an input series of t_i values and the corresponding d_i values,\n", | |
| " compute the summation values that should go into the matrices and\n", | |
| " solve for the 4 unknown variables.\n", | |
| "\n", | |
| " Parameters: time_series -- t_i in increasing values\n", | |
| " data_series -- d_i corresponding to each t_i\n", | |
| "\n", | |
| " Returns: solution -- matrix containing the 4 solutions from solving the linear equations\n", | |
| " \"\"\"\n", | |
| " # compute the data we will put into matrix A\n", | |
| " A_values = []\n", | |
| " for fn in A_fns:\n", | |
| " A_values.append(fn(time_series, data_series))\n", | |
| " # fill the A matrix with data\n", | |
| " A_numerical = Matrix(4,4, A_values)\n", | |
| "\n", | |
| " # compute the data we will put into the b vector\n", | |
| " b_values = []\n", | |
| " for fn in b_fns:\n", | |
| " b_values.append(fn(time_series, data_series))\n", | |
| " # fill the b vector with data\n", | |
| " b_numerical = Matrix(4,1, b_values)\n", | |
| "\n", | |
| " # solve for the unknowns in vector x\n", | |
| " x_numerical = A_numerical.inv() * b_numerical\n", | |
| " #print 'A = \\n', A_numerical\n", | |
| " #print 'b = \\n', b_numerical\n", | |
| " #print 'x = \\n', x_numerical\n", | |
| " return x_numerical" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 21 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Now solve for the 'best fit' solution.\n", | |
| "We will start with the x data from the (x,y) data points and then the y data." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# solve for the best fit in the x dimension\n", | |
| "x_solutions = solve_for_cs(time_series=ts, data_series=xs)\n", | |
| "# solve for the best fit in the y dimension\n", | |
| "y_solutions = solve_for_cs(time_series=ts, data_series=ys)\n", | |
| "\n", | |
| "# place the solutions into control points\n", | |
| "best_fit_control_pts = [(x,y) for x,y in zip(x_solutions, y_solutions)]" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 22 | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "<h2>Now for the test... did it all work?</h2>\n", | |
| "Use the best fit control points to create the best fit bezier curve over top of the 'hand drawn' curve." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [ | |
| "# create a list of 1000 x,y points that traces over the length of our best fit curve\n", | |
| "best_fit_points = [numerical_cubic_bezier_fn(control_pts=best_fit_control_pts, t=t) for t in arange(0.0,1.0,1./1000)]\n", | |
| "# separate the x's and y's so matplotlib can plot them\n", | |
| "best_fit_xs = [x for x,y in best_fit_points]\n", | |
| "best_fit_ys = [y for x,y in best_fit_points]\n", | |
| "\n", | |
| "# plot the original noisy curve\n", | |
| "plot(xs, ys, 'b-')\n", | |
| "hold(True)\n", | |
| "# plot the best fit curve over top\n", | |
| "plot(best_fit_xs, best_fit_ys, 'r-')\n", | |
| "hold(False)\n", | |
| "t=title('best fit cubic bezier curve')\n", | |
| "\n", | |
| "# just for comparisons sake, print the 'clean' control points and then print the best \n", | |
| "# fit control points taken from the noisy data\n", | |
| "print 'Clean Control Points = ',\n", | |
| "print [p0, p1, p2, p3]\n", | |
| "print 'Best Fit Control Points = '\n", | |
| "print best_fit_control_pts" | |
| ], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "stream": "stdout", | |
| "text": [ | |
| "Clean Control Points = [(0.7, 0.2), (2.3, 1.0), (7.2, 0.8), (10.0, 0.0)]\n", | |
| "Best Fit Control Points = \n", | |
| "[(0.680235809341125, 0.199169722342596), (2.30586285000266, 1.03398096289025), (7.22739062250292, 0.753866347077646), (9.99219893606198, 0.00672263149211805)]\n" | |
| ] | |
| }, | |
| { | |
| "output_type": "display_data", | |
| "png": 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xYuTn5+PYsWNIT0/XSEK3Gpo10zom9eCbu3VrLemfPg18/jkVElOxY4d8rpYT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| |
| } | |
| ], | |
| "prompt_number": 23 | |
| }, | |
| { | |
| "cell_type": "code", | |
| "collapsed": false, | |
| "input": [], | |
| "language": "python", | |
| "metadata": {}, | |
| "outputs": [], | |
| "prompt_number": 23 | |
| } | |
| ], | |
| "metadata": {} | |
| } | |
| ] | |
| } |
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