From: Sam Moore Date: Thu, 16 Jan 2014 05:36:01 +0000 (+0800) Subject: Beziars from Iterated Function Systems X-Git-Url: http://git.ucc.asn.au/?p=matches%2FFYP2014.git;a=commitdiff_plain;h=84bd0152f58f92049357d184d1cee7c57722cb83 Beziars from Iterated Function Systems Using Goldman's Iterated Function System to generate a Bezier is horribly inefficient. I guess the point is that it proves that a Bezier is a fractal. The subdivision still looks like it gives a slightly different curve to just sampling P(t), but maybe I aren't using enough points. --- diff --git a/ipython_notebooks/de_Casteljau.ipynb b/ipython_notebooks/de_Casteljau.ipynb index 976a6b6..c7a2805 100644 --- a/ipython_notebooks/de_Casteljau.ipynb +++ b/ipython_notebooks/de_Casteljau.ipynb @@ -13,38 +13,16 @@ "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", + "I think the algorithm developed by Goldman might be worse than just applying the Bezier definition...\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" + "Binomial Coefficients" ] }, { @@ -78,7 +56,7 @@ "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 216 + "prompt_number": 2 }, { "cell_type": "code", @@ -99,12 +77,12 @@ "output_type": "stream", "stream": "stdout", "text": [ - "0.000294923782349\n", - "0.000556945800781\n" + "0.00108003616333\n", + "0.000107049942017\n" ] } ], - "prompt_number": 215 + "prompt_number": 3 }, { "cell_type": "heading", @@ -114,6 +92,20 @@ "Bezier from 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: $\\left(^n_j\\right) = \\frac{n!}{n!(n-j)!} = \\text{nCj}$" + ] + }, { "cell_type": "code", "collapsed": false, @@ -127,8 +119,11 @@ " \"\"\" 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 BezierCurve(P, nPoints):\n", + " return map(lambda t : Bezier(P,t), linspace(0,1,nPoints))\n", " \n", - "def PlotBezier(P,nPoints=50,style='o-'):\n", + "def PlotBezier(P,nPoints=50,style='-'):\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", @@ -139,7 +134,7 @@ "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 266 + "prompt_number": 4 }, { "cell_type": "code", @@ -147,36 +142,58 @@ "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()" + "PlotBezier([(0,0),(0.4,-0.8),(1,1)],nPoints=100,style='-')" ], "language": "python", "metadata": {}, "outputs": [ + { + "output_type": "stream", + "stream": "stderr", + "text": [ + "/usr/lib/pymodules/python2.7/matplotlib/axes.py:4486: UserWarning: No labeled objects found. Use label='...' kwarg on individual plots.\n", + " warnings.warn(\"No labeled objects found. \"\n" + ] + }, { "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 + "prompt_number": 5 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ - "Fractal Beziers" + "De Casteljau Algorithm" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$L P = Q$ and $M P = R$" + "Subdivide the original $n$ control points $\\{P_0 ... P_n\\}$ into $\\{Q_0 ... Q_n\\}$ and $\\{R_0 ... R_n\\}$ (ie: Makes $2n$ points)\n", + "\n", + "Defining $(n+1)\\times(n+1)$ matrices $L$ and $M$ such that $L P = Q$ and $M P = R$\n", + "\n", + "Then it can be shown (subdividing at $t = 1/2$)\n", + "\n", + "$L_{i,j} = \\left(\\frac{\\left(^j_k\\right)}{2^j}\\right)$ and $M_{i,j} = \\left(\\frac{\\left(^{n-j}_{n-k}\\right)}{2^{n-j}}\\right)$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Confusion:** Do $Q$ and $R$ converge to the same curve as applying the definition to produce points along $P(t)$?\n", + "\n", + "It is implied in Goldman, but it doesn't look like it, unless I made a mistake. See below." ] }, { @@ -184,6 +201,7 @@ "collapsed": false, "input": [ "def L(n):\n", + " \"\"\" Gives L of size (n+1)*(n+1) \"\"\"\n", " try:\n", " return L.dynamic[n]\n", " except KeyError:\n", @@ -193,6 +211,7 @@ "L.dynamic = {} \n", "\n", "def M(n):\n", + " \"\"\" Gives M of size (n+1)*(n+1) \"\"\"\n", " try:\n", " return M.dynamic[n]\n", " except KeyError:\n", @@ -205,44 +224,56 @@ "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 296 + "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ - "points = [(0,0), (0.5,1), (1,0)]" + "L(2)" ], "language": "python", "metadata": {}, - "outputs": [], - "prompt_number": 298 + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 7, + "text": [ + "array([[ 1. , 0. , 0. ],\n", + " [ 0.5 , 0.5 , 0. ],\n", + " [ 0.25, 0.5 , 0.25]])" + ] + } + ], + "prompt_number": 7 }, { "cell_type": "code", "collapsed": false, "input": [ "def Casteljau(P, nPoints):\n", + " \"\"\" Approximates the Bezier curve with control points P using the de Casteljau subdivision algorithm \"\"\"\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", + " P = list(Q) + list(R)\n", " return P\n", " " ], "language": "python", "metadata": {}, "outputs": [], - "prompt_number": 329 + "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ - "points = [(0,0), (0.5,1), (1,0)]\n", + "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n", "c = Casteljau(points,50)\n", "PlotBezier(points)\n", - "plot([p[0] for p in c], [p[1] for p in c], 'go-')" + "plot([p[0] for p in c], [p[1] for p in c], 'g-')" ], "language": "python", "metadata": {}, @@ -250,36 +281,225 @@ { "metadata": {}, "output_type": "pyout", - "prompt_number": 332, + "prompt_number": 9, "text": [ - "[]" + "[]" ] }, { "metadata": {}, "output_type": "display_data", - "png": 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Te7Wp2mmvrfuEqtbMcvvAUBdHIoQQzqGV55zZ/adbcs/OLsBofIqj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\nSXIXQggv9P8BcEdRFyfaa+oAAAAASUVORK5CYII=\n", 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"text": [ - "" + "" ] } ], - "prompt_number": 332 + "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ - "So, it sort of works but not giving the same curve as the vanilla method" + "So, it sort of works but not giving the same curve? Or maybe I just need *lots* more points and they will converge." + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Iterated Function Systems" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Defining $L_p = P^{-1} L P$ and $M_p = P^{-1} M P$, then $P L_p = L P = Q$ and $P M_p = M P = R$\n", + "\n", + "\n", + "$\\{L_p, M_p\\}$ is an iterated function system, $\\implies$ Beziers are fractals.\n", + "\n", + "Problem: $P$ needs to be invertable; introduce homogeneous coordinates $\\langle$insert hand waving here$\\rangle$ to \"lift\" it and then it all magically works" ] }, { "cell_type": "code", "collapsed": false, - "input": [], + "input": [ + "def lift(P):\n", + " \"\"\" Lift P to make it square by introducing homogeneous coordinates\"\"\"\n", + " n = len(P)\n", + " result = []\n", + " for i in xrange(len(P)):\n", + " ones = min(n-len(P[i]), max(1, i-1))\n", + " zeroes = max(n-len(P[i])-ones, 0)\n", + " result += [asarray(list(P[i]) + [1. for _ in xrange(ones)] + [0. for _ in xrange(zeroes)])]\n", + " return asarray(result)\n", + "\n", + "def lower(P):\n", + " \"\"\" Lower square P back to list of 2D points \"\"\"\n", + " return [row[0:2] for row in P]\n", + "\n", + "def Lp(P):\n", + " return dot(inv(P), dot(L(len(P)-1), P))\n", + "\n", + "def Mp(P):\n", + " return dot(inv(P), dot(M(len(P)-1), P)) # Typo in Goldman? Reads \"P^-1 * R * P\" instead of \"P^-1 * M * P\"" + ], "language": "python", "metadata": {}, - "outputs": [] + "outputs": [], + "prompt_number": 10 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So, Goldman's method is:\n", + "\n", + "
\n", + "
1. lift the control points\n", + "P from the ambient affine space of dimension $d$ to the\n", + "ambient affine space of dimension $n$;\n", + "
2. generate the corresponding higher dimensional Bezier curve using the\n", + "iterated function system $\\{L_p , M_p \\}$ ; \n", + "
3. project the\n", + "resulting $n$-dimensional Bezier curve orthogonally back\n", + "down to the original dimension $d$.\n", + "
\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def Goldman(P, nPoints):\n", + " while len(P) < nPoints:\n", + " Pl = lift(P)\n", + " LP = Lp(Pl)\n", + " MP = Mp(Pl)\n", + " Q = dot(Pl, LP)\n", + " R = dot(Pl, MP)\n", + " P = lower(Q) + lower(R)\n", + " return P\n", + " " + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 11 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n", + "c = Casteljau(points,50)\n", + "c2 = Goldman(points,50)\n", + "PlotBezier(points)\n", + "plot([p[0] for p in c], [p[1] for p in c], 'g-')\n", + "plot([p[0] for p in c2], [p[1] for p in c2], 'r-')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 12, + "text": [ + "[]" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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ECPX1yy8rYOfinyTxCyGw0umI+GAR5+Z9i9ukj9j4YOtCxd5q1oR586BzZzXw\ne/x4OQeg06nVY1JTy3nH4nakq0cIUUhWWhJHe7ajevY16q2Io0GzkEKvf/01TJigCr61b2+iIAUg\nXT1CiHJSp1ETQv9IJTOiLVqbNuz9eXqh1599Vq2p0ru3euJXWB6D7/gzMzPp378/p06dwsvLi59/\n/pk6deoUaefl5UXt2rWxtrbG1taW+Pj4okHIHb8QZmnnrHdp/MK/OfRUTyI+XozVLQOvu3er5P/i\ni/DSSxU040fckdFn9bz66qu4urry6quvMnXqVC5evMiUKVOKtGvSpAm7du3C2dm5+CAk8QthtlL2\nbeZy72iyGjkT9OsfODg3uPlacjLcfz9ERKhF3o1W3lkAJujqWbZsGYMGDQJg0KBBLFmypNi2ktSF\nsFzuQZ1oeiCVgpo1SGvZhNO7426+5uEBmzdDYiI88oia+VNuduxQc/tFuTP4jt/JyYmLFy8CKrE7\nOzvf/PlWTZs2xdHREWtra4YNG8bQoUOLBmFlxYQJE27+HBkZSWRkpCFhCSEqiKbXs+m1/gR8tYjk\nL6YQ8sT/p33euAFPPgnnzqlF3WvXvsOOSmvNGhg2DPbvBweHctih5YuLiyMuLu7mz5MmTSr/rp6o\nqCjS09OL/P7dd99l0KBBhRK9s7MzmZmZRdqeOXOGhg0bkpGRQVRUFNOnTyc8PLxwENLVI4TF2DPv\nYxo++zKH/9HvX1AAo0fDtm2wahXUr18OBxs0CJyd4aOPymFnlY/R+/j9/PyIi4ujQYMGnDlzhi5d\nupCYmHjH90yaNAkHBwdefrlwbRBJ/EJYlpT9W8iOiSKzSX1Clu+iRm01hqdpqsbPnDnqhr1p0zIe\n6MIFaNFCVY4LCyt74JWM0fv4e/fuzezZswGYPXs2ffr0KdImJyeH7OxsAK5evcqaNWto2bKloYcU\nQpgJ95Yd8dx/Ggr0nGjVmLPH9gBqZs+ECfCvf0F4OOzZU8YDubiou/1nnpEKnuXI4MQ/duxY1q5d\nS7NmzVi/fj1jx44FIC0tjZiYGADS09MJDw+nVatWhIWF0bNnT7p161Y+kQshTKqmoysdNiVx/t52\n5LcNJXHN3JuvDR+uZvl06wYbN5bxQP37Q5s2cORIGXck/iZP7gohymzbxy/j+8ZHHJ/8L9qNnnbz\n97/9BgMHwuzZan11Ub6kOqcQwqQOx/6A48CnONqvCxEzVt8c9N2+XVX3/OIL6NvXxEFWMpL4hRAm\nl340gQvWDAzJAAAVXklEQVTdwrno1YA2K3bfrO+/e7d60GvaNHjiCRMHWYlIrR4hhMk1aBZCk72n\nsMm5xuF7GpOVlgRA69aq22fcOJgxw8RBCkn8QojyVdPRlTZbTpLVwofM1n43n/QNCIC4OJgypYzT\n8vPzYcECNXdUGEQSvxCi3Fnb2hG5cCfJT/bBLrIrB5bPBMDbW1Vh+PJLePttA3O3Xq8eFpg/v3yD\nrkIk8QshKkzE+/M5PW08DQY+w7aPXwFUfZ9Nm+Dnn+HNNw1I/nZ2atGWMWPUA17irsngrhCiwh1e\n8yN1HnmSo0P6EPHBIkAt4XjvvdCrF7zzjgFlnceMgcxMNVe0ipJZPUIIs5aybzO5UV05HdmKzj9t\nQWdtw/nzah3fmBh49927TP5XrkDLlvDVVxAdXWFxmzOZ1SOEMGvuQZ2os/MALrsOse1eX3KvXcHV\nVc32WbECxo+/y24fBweV9H/6qcJirqzkjl8IYVQ5l86zv2tLbG7k0mzDfmq5NuL8ebjvPvV07+TJ\nd3nnr2lVdvkvueMXQliEmo6u3LM1iavu9Um5x4dzJ/bdvPNftQpef/0u7/yraNIvC0n8Qgijs7Gr\nTviKA5zt2o5rYfdwenccLi4q+cfGGpD8xV2RxC+EMAkrnY7Imes59Uw/bLt05WjcLzeT/6pVMGmS\nqSOsvCTxCyFMqvOUufz5+nM49erH/iVf4+ICa9eq57OmTr3LnaWmwokTFRJnZSKJXwhhch1fm87J\nDyfQ8PHh7PruPerVg3Xr4L//henT72JHS5eqxX/1+gqLtTKQxC+EMAtthk4gbfZnNB41nm0fvoSb\nm0r+//mPelC3VIYPV1+//LLC4qwMbEwdgBBC/C2o73McqeOK10MD+f3iBcLfns26ddClC1SvDo8/\nXsIOdDp1lQgPh969VX0IUYTM4xdCmJ2T8Wux6d6DE4P7EPHBQg4dUk/4Tp8O/fqVYgdvvw07dsDy\n5ZV6uqeUbBBCVCop+zZTcG8Xkh6+j8gvV7Fnj6rMMGuWWtTljnJzoV07VcenZUujxGsKkviFEJXO\nmcSdXIvoyOnu7YiYtYEd8Tp69YIlS6BjxxLenJcHtrZGidNUJPELISqljD8PcDG8DWkdWxIxbztr\n1up48kk15TMoyNTRmZaUbBBCVEp1m7agbvwB6sYfZNMDrYi6L59PP1V1fWTKvmFkVo8Qwuw5uXmj\n23GYgk4t2NwziId/3cfFizZ06wabN0PDhqaO0LLIHb8QwiI41m9Mk22J1DmewuaeQTw7NJ+nn1YD\nvhcvlmIHR45UeIyWQhK/EMJi1HJtRJNtiTieSGXz/S0Z+1o+XbtCz56Qk3OHN165ApGRaoqnkMQv\nhLAstVwb0XTrYRyT0tgS05L3p+Xj4wMPPwz5+cW8ycEBPvwQnnlGTfWs4iTxCyEszq3Jf2vPFnw9\nIx+9XlVsKHaSy4AB4OlpQOW3ykemcwohLNaVzHROtGvOpSYNafXzAe7takOvXjBhQjFvOH0aQkLg\n99/B39+osVYEo0/nXLBgAYGBgVhbW5OQkFBsu9jYWPz8/PD19WWqXGmFEOXIwbkB3tuP4HjyDHsG\nBPHrcj2zZ9+hqFvjxqrQ/8SJxgzT7Bh8x5+YmIhOp2PYsGF88MEHhISEFGlTUFBA8+bNWbduHW5u\nbrRp04a5c+fi/48rrdzxCyHKIvt8GqfaNudCsA8N3ttFRKSOmTOLKe2g16uRYAcHo8dZ3ox+x+/n\n50ezZs3u2CY+Ph4fHx+8vLywtbVlwIABLF261NBDCiHEbdVybYTH1gPU23WEM291YvEvegYNgj/+\nuE1jna5SJP2yqNAHuFJTU/G4pSyqu7s7O4qZTjXxlo9ekZGRREZGVmRoQohKxrGBJ/mbd2PTLpjU\nGlF8++1v9O6tuvN9fEwdXfmIi4sjLi6uzPu5Y+KPiooiPT29yO8nT55Mr169Sty51V2UQ51Yxfvc\nhBBl59K4OQW/x2PdIZTTNXoxYcJyevSAbdvA1dXU0ZXdP2+KJxm4MPEdE//atWsN2unf3NzcSE5O\nvvlzcnIy7u7uZdqnEELcST3vIArittA0vANWgx+mb98F9OmjVvOqXv02b7h+HW7cAEdHo8dqKuUy\nj7+4wYXQ0FCOHTvGyZMnyc3NZf78+fTu3bs8DimEEMVq6N8G3foNeH+7mJjqQ2jYEIYMKWaO/wcf\nwKhRRo/RlAxO/IsXL8bDw4Pt27cTExNDjx49AEhLSyMmJgYAGxsbPvvsM6KjowkICKB///5FZvQI\nIURFcA/qRO7KX2n24Xe80OE1jh9XMzmLeOEFNRCwerXRYzQVeYBLCFGpHV7zI64PPcGh96fy1LRX\nePvt26zdu3q1eux3/36LmvEjC7EIIUQx9sz/BLchY9jz4Swee2MQixap9dgLefJJcHaGjz82SYyG\nkMQvhBB3EP/Vm3i9OplNk5cw8p1e/P47+Pre0uDCBQgMhA0bLKacgyR+IYQowZbJI/B6/78seWkj\nn8zpyLZt4OJyS4OUFHBzg7uYim5KkviFEKIUNr7cl8ZzlvPVwwn8cbgFq1db7prskviFEKKU4p6K\npP6GeMYFHqdRk0Z8/rmpIzKMJH4hhCglTa/n9x6B1EjL4OmC0zw/uibDh5s6qrsniV8IIe5Cfu51\nEtp5ctW+FgOOHmX+fB1FSoTl5EDNmqYIr1SMXp1TCCEsmY1ddQJ/20/d1HRmtO3MgAHw55+3NLh8\nGfz84JayM5WFJH4hRJVl71SPBuv/IHhHPO90fJTevVW+B6B2bRg6FEaMuMN6jpZJEr8Qokpz9fJH\ntyqWnuvm06/JOB5/HAoK/nrxtdfg1CmYP9+kMZY3SfxCiCrP8557yZw3i5EbpuCU+TVvvvnXC3Z2\nah3HMWPUA16VhCR+IYQAAno8SdLHk3h/7wjiFq5n4cK/XggLg/794dVXTRpfeZJZPUIIcYtNYwfi\nNvMXomwSWb62CYGBwJUrkJ5udkt5yXROIYQoJ3EP3UP1fX/ytHUyW3c4UKeOqSO6PUn8QghRTgry\nctnZ3pPU3FrMbJzIsmU6dGbYMS7z+IUQopxY29oRuGY3/lkphKfez1tvmTqi8iWJXwghbsPBuQF1\n1mzi8ZPrODX/JZYtu+VFTbtlzqflkcQvhBDFaOgXypUFP/L+6Y/59MWZHDny1wvvvac2CyV9/EII\nUYL4Gf/G45V36e+1mZVb2+NwMRlCQmDTJpMu2iKDu0IIUYHiRvXGacE6pnU+zQ/zXbH64nOYO1cl\nfxON/EriF0KICqTp9WyO9OFcaj5nXjzJyOeBzp3h0UfhuedMEpMkfiGEqGDXLmdyPKgxC+06EvPD\natrWOqySf0ICeHgYPR6ZzimEEBWsRm1nXGPjGHZ2HZ8+9SYX6vnDggVQt66pQ7srcscvhBB36cCy\nb6g38Flear+U79f0MtnDXXLHL4QQRtKi9zMkvjqEiQl9eWd8kqnDuWtyxy+EEAZa2+se2H0S7ZtU\nunWvbvTjyx2/EEIYWZdFW6jlAIdGdSY11dTRlJ4kfiGEMJCNXXWard3OQ+cTmNbvFfIzLkJEhCrj\nbMYk8QshRBk4e/hy+YeZvLH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+ "text": [ + "" + ] + } + ], + "prompt_number": 12 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The algorithm works, I guess, but is it useful? It will be a lot slower than de Casteljau, because it does a lot more matrix manipulation stuff, but it gives no more accuracy on each iteration." + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Time Performance of Casteljau, Goldman, and Basic Bezier creation" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from time import time\n", + "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n", + "costs = {BezierCurve : [], Casteljau : [], Goldman : []}\n", + "stddev = {}\n", + "nAverages = 10\n", + "depth=8\n", + "for method in costs:\n", + "\n", + " stddev.update({method : []})\n", + " for n in xrange(1,depth):\n", + " #display(\"depth %d for %s\" % (n, str(method)))\n", + " c = []\n", + " for _ in xrange(nAverages):\n", + " t0 = time()\n", + " p = method(points, nPoints=len(points) * 2.**n)\n", + " c += [time()-t0]\n", + " costs[method] += [mean(c)]\n", + " stddev[method] += [var(c)**0.5]" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 17 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "title(\"Time to produce $n$-dimensional Bezier Curve\")\n", + "xlabel(\"$n$\")\n", + "ylabel(\"$t$ (s)\")\n", + "x = [len(points) * 2.**n for n in xrange(1,depth)]\n", + "for method in costs:\n", + " errorbar(x,costs[method],yerr=stddev[method],label =str(method).split(\" \")[1])\n", + "legend(loc=\"upper left\")" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 18, + "text": [ + "" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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5GabrBfjxxx8JCgrCwcEBb29v5syZY1iWkpKCpaUlq1atwsfHBxcXF8Osi0LU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\nkFfvexV/Z3+5XkOIWmSyRGJtbc3SpUv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+ "text": [ + "" + ] + } + ], + "prompt_number": 18 } ], "metadata": {} diff --git a/ipython_notebooks/fractals_basic.ipynb b/ipython_notebooks/fractals_basic.ipynb index 38b7eee..7ce2dea 100644 --- a/ipython_notebooks/fractals_basic.ipynb +++ b/ipython_notebooks/fractals_basic.ipynb @@ -169,7 +169,7 @@ " normal = asarray([-s[1], s[0]])\n", " normal = normal / sum(normal*normal)**0.5\n", " # Calculate the three points \n", - " # Are these are the maps that form the iterated function system?\n", + " # Are these are the maps that form the iterated function system? I don't think so.\n", " # \"Notice that in each case the entire fractal can be recovered from a single point.\" Goldman\n", " # But for this I need *two* points :S\n", " \n",