De'Casteljau and Splines
[matches/FYP2014.git] / ipython_notebooks / de_Casteljau.ipynb
index c7a2805..1ebf62e 100644 (file)
       }
      ],
      "prompt_number": 18
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Approximating a Circle using Cubic Beziers"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "k = 4. * (2.**0.5 - 1.) / 3.\n",
+      "points = [(0,1), (k, 1), (1, k), (1,0)]\n",
+      "c = BezierCurve(points, 50) # This appears to be more accurate than using the de Casteljau algorithm :S\n",
+      "c += list(array([-1,1]) * c)\n",
+      "c += list(array([1,-1]) * c)\n",
+      "scatter([p[0] for p in c], [p[1] for p in c])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 125,
+       "text": [
+        "<matplotlib.collections.PathCollection at 0x8eb2650>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAX4AAAD9CAYAAAC7iRw+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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8O23qzVOlveowp/D5s1bjrzZdBn5dk+hRImSNG3aVNtgilLAFC5cmK5dOwL9gBrI2fF9Wbt2o7bB\n8iBV+HPYhQvnkItIHQBeAr4hNLSKppkUpaCqU6c2BsMx5P4WvwDfkJpqV2v3/Ikq/DnoypUrXLly\nFbmR+nzkzMItvP66Gs2jKDmhT58+eDy/IOfKfA38gl4fwqpVajLX76nCn4Nu3ryJEL5AVeQKgh/j\n61uBSpUqaZxMUQqmoKAgDAYXcAwoDRTF6UwjJiZG42R5iyr8Oejo0WM4HAbkJecZ4Cgezw1CQ0M1\nTqYoBVdo6GPABOTSKJfxeMqze/dBjVPlLarw56DNm3cDvZF9/KuAUZQpUxF/f39tgylKAda4cQPk\nqJ4TwAygLdHRh7QNlceowp+DUlISkGvuZwCRwCDKlg3UNJOiFHQVKwaj030LfA64gTk4nS6NU+Ut\nagZRDjp8+ATwI3L9fRPQGputnrahFKWAa9HCBnwGHEaWuGGkpVUgLi6OsmXLapotr1At/hyUlpaE\nHMNvAcpjMBRR3TyKksPcbjd+fqWRs3etQCgGgy93797VNlgeku3Cv3nzZqpWrUpISAiTJ0++7zlD\nhw4lJCSEsLAwjhw5kt2nzBdcLhdgBt4EnMB83O4ItcWiouSwmjVrkp5+DegCZAJrycxM0ThV3pKt\nwu92uxk8eDCbN2/m1KlTfPfdd5w+ffoP50RERHDhwgXOnz/P3LlzGTRoULYC5xexsbHodBbgJNAC\nWI/VWjfrA0FRlJyk0zmQrf2WwHQslhZe0+j8/5Gtwh8dHU2VKlWoWLEiJpOJnj17sn79H/e63LBh\nA3379gWgQYMG3Llzh5s3b2bnafOFYsWKkZGRDGwDmgO7sduPUapUKY2TKUrB5uvri8fjQQ7pbALc\nxW6PxM/PT+NkeUe2bu7GxcURHBx877hcuXLs37//v54TGxtLYOAfR7eMGTPm3tc2mw2bzZadaJo7\nduwYoEPO1i2E3Ai6HPHx8ZrmUpSCzmAwIIQOuTxzdUAATdi2bRtdu3bVNlw2RUZGEhkZme3HyVbh\n///dK/bP60Xf7+d+X/gLghs3biBn6xbK+hNfoATXr1/XLpSieAHZneoAKmb9iQ6oTGJiomaZHpY/\nN4rDw8Mf6HGy1dVTtmzZP0yFjomJ+cuGB38+JzY21iuGVD311FNAInK7xTjgCyCO+vXra5pLUQo6\no9GIXh8ADARikEOq19CqVSttg+Uh2Sr89erV4/z581y5cgWHw8GKFSvo3LnzH87p3LkzixcvBmDf\nvn0UKVKuJZn5AAAgAElEQVTkL908BVFycjI+Pv7AZqAu8D2+vjXUmiGKksNSUlLQ6ezIyVsNgPcw\nm+tisVg0TpZ3ZKurx2g0MnPmTNq2bYvb7aZ///5Uq1aNOXPmADBgwAA6dOhAREQEVapUwc/Pj6+/\n/vqhBM/rihUrhtudAiwCgoFU7PbK6gaTouQwq9WK7F3+ELkBuwOP53E1sOJ31J67OcjPryTp6T5A\nR2APBoOLmTPfZuDAgVpHU5QC68SJE9SpY8PpNAGdgEPo9Re5fPkY5cuX1zreQ6X23M2DhHABc4Ew\n4DPc7jYF4gaTouRlycnJ6PVlkX37tYGPsFiKk5qaqnGyvEMV/hwkl4edA3RAjixYzLFjZ7UNpSgF\n3JUrV8jMvAzsBZ4GjiNEClWqqJ3v/kMV/hxUrlwwkIKcRDIc+IBjx85pG0pRCrjo6CPAq8hduBoA\nW7FaLZjNZm2D5SGq8OegKlWC0ekqArHIHYE8xMffLJD3MxQlrzh//ixyqZSdwG/AK5QuXfCHkP8v\nVOHPQd27d0W2Op4COgMzSUtL5+xZ1d2jKDklMnIvkAA0Ap4DBtOly1Pahspj1Hr8OchiseDrW5q0\ntMHIFToX4nDU5vbt21pHU5QCSQhBRkY6sB65Hn8qZrOJ0qVLa5wsb1Et/hz0+OOPYzKlAXuASsBE\nPB4PCxYs1TiZohRMy5cvR68vDvQHSgJpuFw/0LFjR42T5S2q8Ocgs9lMzZphyJbHICAemM4vv/yq\nbTBFKaAOHDiIy/USchjnEGAZvr4+VKpUSdtgeYwq/DmsYcNa6PXFgGhgMbCNCxfOEBcXp3EyRSlY\n3G43K1euQ47f/wA4ADzNo48+pm2wPEgV/hz2/vsj8Hi2ASFAVeAoBoONrVu3apxMUQqWixcvcvu2\nA7nHdRWgFjrdWEaOfEPjZHmPKvw5zN/fH5PJDcwDVgB7ycy8zfffb9A4maIULLt37yYj4y4wA7kW\n/xys1uJq4tZ9qLV6csHAgUOYMycCGIbs79+D2fwbV6+eJSgoSON0ilIwBAVV5ubNyshVObsDawkJ\nucmZM4fR6wtmG1et1ZOH9e7dHatVDxwBgoAonE5/9u3bp3EyRSkYkpKSiI+/DnyD3GT9EDqdgz59\nniuwRT871DuSC2rWrInRmAzYgJHAcoTw8MYbwwr0lY6i5JaBA98BKiBv6vYFXsNkOkHr1q21DZZH\nqcKfC4oUKcLw4YPR6d4BSgBfAz9x/XqMV2w8ryg5SQjB1q07EWIOcAcoDXSgQYMwGjRooHG6vEkV\n/lxis9mwWv2QY/mjgSSEMPPSS69rnExR8rdPP51GcnImsB+5REoqZnMT2rVTWy3+HXVzNxe1avU0\nO3YcRE4uOQjMR6/vTmrqXaxWq8bpFCV/Cg5+nNjYD5HdqJWAmxQvbicm5lSB/71SN3fzgddffxmr\nNRh4HTm6pxEej+DNN0dqnExR8qfvvltOXNwNwBc4iiz+9ejevXOBL/rZoVr8uSg9PZ3Q0DrExtYB\nmiJ353oKk2kecXEXKVmypMYJFSV/CQysQnz8a8A05J4Xt/HxmcXx49GEhIRonC7nqRZ/PuDr68uX\nX36Cj88+5NDOd4BPcTpNDB/+gcbpFCV/WbFiJfHxN4B/AcuBi8BPvPZaH68o+tmhWvy57O7du1Su\nXIPbt4cjt2ScD2zGak3ip5+W07BhQ40TKkrel5GRQdGiQWRktAeSganAJSyWvuzdu5k6deponDB3\nqBZ/PhEQEMAPP6xEp/sQaAmcASKw2x9l2LBRuN1ujRMqSt43efJnZGTogEXIdXk6oNP1Y/jwgV5T\n9LNDFX4NNGjQgFKligETgLXAXeAohw6lMm3aF9qGU5Q8bv/+/UyePAuwAhHA58BaLBY3/fr10zZc\nPqG6ejTy66+/Uq+eDbfbAqQjF5bSU7x4OKdORVGqVCmNEypK3uN0OrHZWhMVVQm5x8UzgAdI5Ntv\nF9G7d08t4+U61dWTz9SqVYv27duh1/cCbgB9gEgSE0vSvv1zGqdTlLxp5MiPOHjwDnI4dE3gMjCF\nEiVKeV3Rzw7V4tfQtWvXqFmzIXfvVgQEkAH8BATxww/fq+3iFOV3Ll26xGOPNSYjYyPwBbAPqITR\nuI/Nm7+nZcuWGifMfarFnw+VL1+eBQtmYLHcAj4CfgFSAD3du7/EgQMHtA2oKHlEUlISDRrYyMgI\nAC4hV+H8Gr3ewGuv9fXKop8dqsWvMZfLhc3WkaioFIRojtysZSiQQt26m9m9e7uagah4NSEEgwa9\nzvz5p3C7xwLPIbtGYyhZcj/Hj0cTGBiocUpt5HqLPzExkdatW/Poo4/Spk0b7ty5c9/zKlasSM2a\nNalduzb169d/0KcrsIxGIzt3/kDJknHIvv65wFtAMr/+eod27Z7x2g9FRQEYPXocixZtxe3OAJoB\nkYA/ev1aDhz42WuLfnY8cOGfNGkSrVu35ty5c7Rs2ZJJkybd9zydTkdkZCRHjhwhOjr6gYMWZCaT\nialTJ2KxbAXOA6OBxbjda9m79yBz587VOKGiaOPo0aNMmvQZmZkbABfQD9iH2RxBv34DqFChgsYJ\n8ynxgEJDQ8WNGzeEEEJcv35dhIaG3ve8ihUrioSEhH98rGzEKFCmTp0qjMbSAoYIOC/AJSBQWCxl\nxMKFX2sdT1Fy1a+//ir8/EoIKCLgooC7Aj4QOt3jolu3Z4Xb7dY6ouYetHY+cB9/0aJFSUpK+s+H\nB8WKFbt3/HuPPPIIAQEBGAwGBgwYwKuvvvqXc3Q6HaNHj753bLPZsNlsDxIrX3M4HNSq1Zhz52rh\ndncDlgHXgdH4+fVi794IwsLCNE6pKDkvPj4em60jp08/B6QCW4APgTMULjyF48cPUL58eW1DaiAy\nMpLIyMh7x+Hh4Q/UFfyPhb9169bcuHHjL38+fvx4+vbt+4dCX6xYMRITE/9y7vXr1yldujS3bt2i\ndevWzJgxg6ZNm/4xhBff3P2zpKQkbLb2HDuWArQHxiAXn+qAn5+D3bu3Urt2bU0zKkpOunPnDtWr\n1+PGDX+EeA0YiJzguJzChX9j377NVKtWTeOUecOD1k7jP31z27Ztf/u9wMBAbty4QVBQENevX//b\nmaalS5cGoGTJknTr1o3o6Oi/FH7l/xQtWpTFi+fQqFEr0tNrAQeQy80OIi3NyrPP9mXLljVq9UGl\nQLp79y49evQmPr4qQrwNvAAEAVXx9U1kypSPVNF/CB745m7nzp355ptvAPjmm2/o2rXrX85JT08n\nJSUFgLS0NLZu3UqNGjUe9Cm9RlhYGJs3f4+f3yjkCJ9ewHtAES5fNlOvXhMuX76sbUhFecgyMjJo\n2PApduxIwu0ugVzEcB5yLZ5eTJs2nFdeeVnbkAXFg95UuH37tmjZsqUICQkRrVu3FklJSUIIIeLi\n4kSHDh2EEEJcvHhRhIWFibCwMPHYY4+JCRMm3PexshGjQFuw4Gvh6xsiYJuAHwSUzfr/WyIkpIY4\nduyY1hEV5aG4deuWaNPmaWEw1BRwWUBJAXME/CKs1nbi+edf0TpinvSgtVNN4Mrj5s1bwOuvj8Ll\nKo/cVq47crhnFP7+v3L4cJTq9lHyNbvdzuOP1+fq1Yq43SnIcfpHgFHAfgYN6sv06Z9gNpu1jJkn\nqSUbCqhXX+3Phx++ja+vBygGLAFmAZ+Smvo8TZq0YuPGjdqGVJQHdOrUKerXb87ly+B2LwGuAmOB\nVHx8itCiRWNmzZquiv5Dpgp/PvDhh/8mPLwPRmNvZJ/n90AtQE98fGN69HiV3bt3axtSUf5H8fHx\nNG7cihMnGiCEBQhAtvZPotN1oUePADZuXKFtyAJKFf58QKfTMWzYW4wbNwI/v0QgHrnV3BIgHLt9\nGDZbB15//W21g5eSLyxY8DUVKz7KnTvVgE+QK9O+CRzFas2kTRsb33wzBz8/P22DFlCq8OcjI0a8\nw9Spb+Pr+xZy56GfgBDgNh5PHxYtOsibbw4jIyND26CK8jc8Hg9Lly5l8OAPsNvDASdgAXYiNyTq\ny+DBj7Nu3TJNcxZ06uZuPrR+/Xp69RqI3T4cuAUsBKKA8+j1L1G8uIXIyAiqV6+ubVBF+Z3k5GSe\neqoTR48exeUaDLwPNAGqAU/i6zufgQPb8dlnE7UNmo+om7tepEuXLkREfEdIyGLgZ2TRrwycwOOx\nkZDwHi1aPM3GjRvVB6qSJ0RHR9OuXVeOHq2IyzUWOIVs6UcCOvz8xjNr1ttMmTJB05zeQrX487Fr\n165Rp05jkpOfxOk0ATuAXUAwEIC/f3U6d36CpUvnodPptA2reK05c+bzzjujsdt9EWIW0Ai5vHIx\n9PoKWCzrWbduGa1bt9Y4af6jWvxeqHz58pw6dYjXXiuF2bwTWfRDgfVAVVJTf2H58gjq1WvO5s2b\ntQ2reJ2zZ8/SuXNPBg16g/T0SIRoify36QvswmRKo3nzaxw4sEsV/VymWvwFgBCCAQPeZMmSVWRk\nFAXuAD8AdQAb0ABf38UsWvQFzz77LHq9+rxXco4QghMnTtCkSWuSkwcD45E3bhOBNsBdLBaoXr0M\nu3ZF4O/vr2ne/OxBa6cq/AXIyZMnsdnakZAwBDk0bhvwMnAc+AG9fjhFivgSEbGGBg0aaJpVKZiu\nX79OmzbdOH36KG53V+A7oB7QCTnz/Gd8fP7FihXf0LFjR4zGf1wnUvkvVFePwmOPPUZU1E9UrboK\neTk9EDnZKxBw4fG0IzHxS1q06MCkSZO4du2apnmVgiMtLY3Zs2fTqFFrTp+24XZ/CpiyvrsW2AT4\nUaLEq6xfv4IuXbqooq8h1eIvoCIiIujevT/p6eORY6U/BFYjb6oFYzI1xmqNJDp6F6GhoZpmVfK3\ntLQ06tRpSkxMeez2Hcj9IzxAXaA/8Di+vpMZOrQjEyeO1TRrQaNa/MofdOjQgXXrFtO8+Vr0+veA\n+ciifxFIxun8iuTkN6levQGVKtUgKipK28BKvhMbG0vDhq0oUqQE58+XxG5fixxW/DNyDf1dGAyL\nqVZtGp988ioTJoRrG1i5R7X4vcD7749l+vR5ZGY+htt9CHmz7TVkN9B84FUslv4MHz6YDh060LBh\nQ03zKnnb5cuX2bBhAxMmfM7t231wuwWQDEwD9iH782vg75/AE0+UZ+vWdapbJ4eom7vKPzp8+DAz\nZnzJ8uVRZGRsAATQAxiAvBfQEqPRitl8hC++GEffvi+qX1blDzweDwcPHqRly6dxONricKwH7iKX\nUG4PrAIexWx+nbCwW0ycOAabzYbBYNA0d0H2wLXzgVbxf8jySIwCz+PxiA8+CBcBAaUF+AoYJsAj\nwCHgMQE7BZwSYBZ6vUk8++wLwm63ax1byQMmTPhEmM2+AgoLmCsgLevfUJwAIWCV0OmKCIulqOjW\n7XmRnJysdWSv8KC1U7X4vdSkSVP4+ONZZGQ8g8ezCygHrEFeCViBm5jNLxIcHEujRg0YNuwNatas\nqWlmJXfdunWLjz+ezP79B/j11xgcjkjgOWA6cvbtZGAWOt1z+Pruo2nTIH78cZWaJ5KLVFeP8j/b\nsWMHERERzJw5F4djLVAfCAcOI1dLjAL6Af3x9f2E8eM/oEaNGjRv3lx1AxVghw4d4uTJk4wcGc7t\n2x1wOi8iJwKOAD5A9uN/C9zFx6cVXbs2pnPnTvTo0UN16+QyVfiVB7Zp0yb69RvMrVtxCBGMEHuQ\nY/+/QK7/sx74GJNpMT4+foSFBbJ27VICAgLUzkgFhMfj4c6dO0ycOJVZsxYjRCB2u1xaQbbsDwPL\nARfwNDrdbvz9/Rk1ajijRg3XMrpXU4Vfybbbt29Tu3ZjkpIeIS3NiBC/IH/xqyMv788AX6LTtUCn\ni8Zg0DNu3DhGjHhH09xK9mzbto1nn+2N3W7H5TIDF5BLfqxHdv+lAk0BE35+Ieh029ix4wfq16+v\nYWoF1Dh+5SEoXrw4J08eYO7cF3nxxeJYrQHANWAFMBHoDRgQoiUezzs4nWd5//2pFC1amipVavPj\njz9qml/5/3f69GmeeOIpihcPpn37Z0hJWYPLtRK5pWcxoC3wC/ApcAiLJYiGDa3MmNGKkycPqqKf\nz6kWv3JfQgjmzl3AV18t5dKlq6Sm1sHjWYXc+KU58vK/CzAcOTO4HWbzC4SEVKFUqUDCw4fRtGlT\nDV+B8mdXr17l7bc/4NKlq5w9e4rMzHCEKAJ8DkQDMciF/TYh19f5HJPpY0JCHqVdOxsTJozGx8dH\nw1eg/Jnq6lFyTHx8fNbCWydxONKRIzuWIft7WyIXguuLXIQrBmiDj8+7NGrUgMKFizF48Eu0atVK\ns/ze7Pz584wd+ynXr8cTHb2P9PTXcbvNyOU7DgLnkN04J4CSwFxgKD4+fhQq5MfmzWupW7eudi9A\n+Ueq8Cs5SghBYmIiR48epUuXnuh0TUhNPYEQJZH3AYzAC0Bt4F1gEnJ10J5YLB/SpEldzOZCPPNM\nW15++SW1MUwO2rVrF198sYD09DR27/4Zu/1NPJ4byF2vfkKOynkJuWqrCRgFzMPf34bHs5fx49+n\nV6/ulCxZUg3NzONU4VdyTWxsLFFRUfz661GmT1+I3f46cB45BPRXoDhyWGgS8qbwd8hlIv6Nr+9k\nqlULwGDwpXbt6nzyyVgKFy6s1UspENxuN598MpX167djMLg4fPg4GRnhyC6b0sAcZAt/DvLD2AN0\nRC6z8DR+fqto164qzz7biccff5waNWpo9VKU/5Eq/Iomfv75ZzZujODq1cv88MNe7PbxwG3gY+Ri\nXTWQawLNBTYjuxaeAJbj47OKQoV2YjKZKVw4gM8//5innnoKh8OBn5+fVi8pT3M4HLjdbmJiYujX\nbwgXL17CYjETHx+A3f4+8BEwBNmiH4f8u5iGLPK1gA5AU6zWGdSqZaBBg/o88URtevXqpa7C8iFV\n+BXNrVu3jnnzlpOaepfo6ANkZMwEzMAbyDkB3ZGjhOoibxJ7kPsDzwRMmEwv4vGko9PpadCgGdOn\nj+fatWtUqlSJ2rVra/SqtHXlyhWOHDlCUFAQixevZP782Xg8AoPBF7d7NB5PO+T7eR4oA3RFrsHU\nC/kh2wB5tfUIVusIqlYtTIkSQbRp05i33x6qJlzlc6rwK3nKpk2bmDBhJmlp6Zw+fZjMzHEIEQr8\nG2gFfAK4kcv4/gg8hpwZ6gEmYTAMQIi1+Ps3xeU6xHPPtcdud+LxCAYP7kezZs1ISkqiaNGi+b4f\nWghBUlIShQoV4tq1a3z88RRu375LSEhp5sxZjNH4JHb7PoQIxuXajhxmORo4kPUIRZD99cHIsfev\nArMBDxbL61StWg2j0UyvXp14++2hqmVfgKjCr+RZx48fZ/jwcOLjb3PzZiy3b9cjM7M7sBQ5Cmg/\nckrJi0AYMAzZgm0FXAUSgCrIK4dgfHw+xGz2kJnpxmLxYdasqZw8eZrMTCfPP9+DmjVr8ttvv1G8\nePE802XkcDi4fv06pUqV4s6dO8yePYfk5DSefLIe778/gZiYywjhwmi0kpk5FI+nInLV1Ehkq70X\ncmz9S8ilNF5C3qw1It+vTcAYDIaTWK0zCA2thY+PmREjBtKlS5dcf71K7sj1wr9q1SrGjBnDmTNn\nOHDgAHXq1LnveZs3b+att97C7XbzyiuvMHLkyL+GUIXfa6SmpvLhh+P49dfTWK16du7cR0bGm+j1\nF/F4ViO7J0oCi5H3BfZk/WQz4D2gHfIDYzawF3lD+Wl0un4IURKLZQYWixGHw4DbncLIkcPYv/8Y\nN27c4umnW/Liiz3YuHEjJpOJnj174na7OXHiBOXKlaN69ercuXOH+Ph4KlSogI+PD1evXiUlJYXq\n1asjhODq1av4+voSFBREXFwcJ0+epHz58lSqVImlS5cSHx9Ps2bNuHLlKtOnL8RoNPLss60YP34q\nDocejycZk8kXu/05XK4y6HSfAKMQYgTyiicJuUdCatb7kA7okENlE4AFyCulMPT6Ung8HbBalxMW\n5o+vb1FKly7BuHHvU7FixVz421S0luuF/8yZM+j1egYMGMBnn31238LvdrsJDQ1l+/btlC1bliee\neILvvvuOatWqPZTwSv63c+dO1q37kUKFrKxbt4WrV41AGVJTNwFjkUNDDwBtkN0Z5ZAjVL5GdhGB\nvGn5JdAY+aEwDTiEXHqgHnLPgbZYLB/jch1Bp3sBvT4Vs3kzLpcDs7k2DscpGjWqx549P2MylcBs\nzsTf359r1y4DRqzWQpQtG8RvvyXgdqdTv35dDh06gskURmbmCYoUCSA5uSKZmbXQ6+ej0/nhcHwJ\npAGvIGc/Pw28jpzwNg9ZwM1AJrLlPgG4gbwfAhCCnCD3GrAPna4NVmtdDIZCGI2HGDy4PwkJyTRs\nWIcXX3xRdeF4Ic3W47fZbOLQoUP3/V5UVJRo27btveOJEyeKiRMn/uW8hxBDKQAyMzPFhg0bxLJl\ny8QPP/wgihUrKyyWEsLHp7AwmYoKWCngewElBMzOWgf+hoCiAi5lHW8X0CTrayGgq4AlWV/fEmDJ\n2oNACHhXQI+srxOyHndd1vFKAYUEJAlwCXhJQM2sn00QYBVwIOvcuQLq/u5xm/zucS4JKP27PCMF\nhP/uOEjArqyvjwvwEzBLwBZhsdQQ/v6BwmIpLnx8/MXs2XPEmjVrxIoVK8StW7e0/utS8oAHrZ05\nurZuXFwcwcHB947LlSvH/v3773vumDFj7n1ts9mw2Ww5GU3Jg8xmM506dbp3HB9/lVu3blG8eHE2\nb97MxImz8Hg8hIQ8zfffj8Vg2IHDsRe324DLdQ1IRLao22c9wl3k1cKwrGM7smX9H9WB+KyviyOv\nDjKzjrsj+9EFYACGItcq0gEZgG/W+SBvSFfN+h6AX9ZzAZQCUpBXKzWQs2R7I4e0lsXHpyRud2es\n1pYIcZp69ZpiMGwlKeku//pXX959901u375NkSJF1HIJCpGRkURGRmb7cf6x8Ldu3ZobN2785c8n\nTJjwh1/Qv/O/XHr+vvArCoDBYCAoKAiATp06/eHf3PDhxzh16hSPPjqK48dPMHbsWzidDmy2xqxe\nvRSz+TQOxymEyMTlWo/LdQmTaTJClMr6kEhDdiV1y3rEC8ibph9mHe9GFvJCWcfbkbNcQX5IOICN\nyP1lywPrss4Jw2AwIcQgPJ47QAYmkw693obF8gQOx1G6dOnM4cMfkZaWSo8e3Rg4sB+HDx+mVKlS\n2Gy2v/zeBAYGPqy3VMnn/twoDg9/sA3s/7Hwb9u27YEe9D/Kli1LTEzMveOYmBjKlSuXrcdUFICa\nNWve2xGsTp069O3b5973Jk6M4+jRo5QpU4ZSpUoRHj6J337bRMeOQzl58hyLFj2B0WjihRe6s2zZ\nKjIz1+FyJdKsWXP27HkWszkUp/MkLpcRh+NxoCg63XGKFCmM2/0kHk8ilStX5+rVATidQ3G5Ennj\njUGsWDGYxMSbNGny/9q7v5Cm/jcO4O9lQgXO/tDWctJq/hmuuWnm6EJIctE0h915ZxEiGHXxhYiu\nIzGouy4GET+M700FrYIa1C+UoBoDd5dSEQX700atAinIhOd39ZO+X52eztaZZ+f9unL6kc/z+Mw3\n8+yc40GMjv4Hf/8dwfr1Vfjrr//CYrHg5cuX2L17N1paWpb009jYqNWPjqj40zm7u7tx+fLlZW/k\ntLCwgObmZjx58gQ7d+5EZ2cn39ylNeXHjx9IJpOwWCwwm82YnZ1FJpOB2+2G2WxGOBzG3Nwcjh8/\njm3btmF6ehobN27Evn378PPnT6RSKVitVtTU1Ky+GVGJaX5WTyQSwZkzZ/Dp0yfU1taira0N0WgU\nmUwGw8PDi/dmj0aji6dznjx5EufPny9Z8URERsYLuIiIDIb/gYuIiBRh8BMRGQyDn4jIYBj8REQG\nw+AnIjIYBj8RkcEw+ImIDIbBT0RkMAx+IiKDYfATERkMg5+IyGAY/EREBsPgJyIyGAY/EZHBMPiJ\niAyGwU9EZDAMfiIig2HwExEZDIOfiMhgGPxERAbD4CciMhgGPxGRwTD4iYgMhsFPRGQwDH4iIoNh\n8BMRGQyDn4jIYBj8Gpiamip3CX9MJfcGsD+9q/T+1FId/Ldv34bb7UZVVRUSiUTBdQ6HA62trWhr\na0NnZ6fa7XStkp98ldwbwP70rtL7U2u92m/0eDyIRCIYGRlZcZ3JZMLU1BS2bt2qdisiIioh1cHv\ncrkUrxURtdsQEVGJmaTIVO7u7saVK1fQ3t6+7Nf37NmD2tpaVFVVYWRkBMPDw0uLMJmKKYGIyLDU\nRPiKr/gDgQCy2eySz4+NjaG/v1/RBs+ePYPNZsPHjx8RCATgcrnQ1dX1jzX8i4CISDsrBv/jx4+L\n3sBmswEAtm/fjmPHjiEejy8JfiIi0k5JTucs9Ir9+/fvmJubAwB8+/YNjx49gsfjKcWWRESkkurg\nj0QiqK+vRywWQ19fH4LBIAAgk8mgr68PAJDNZtHV1QWfzwe/34+jR4/i8OHDpamciIjUkTK4deuW\ntLS0yLp162R6errgul27donH4xGfzyf79+/XsMLiKO0vGo1Kc3OzNDQ0yPj4uIYVFiefz0tPT480\nNjZKIBCQL1++LLtOT/NTMovTp09LQ0ODtLa2SiKR0LjC4qzW3+TkpJjNZvH5fOLz+eTChQtlqFKd\nEydOiMVikb179xZco+fZrdafmtmVJfhnZ2fl1atXcvDgwRWD0eFwSD6f17Cy0lDS38LCgjidTnn3\n7p3Mz8+L1+uVmZkZjStV5+zZs3Lp0iURERkfH5dz584tu04v81MyiwcPHkgwGBQRkVgsJn6/vxyl\nqqKkv8nJSenv7y9ThcV5+vSpJBKJgsGo59mJrN6fmtmV5ZYNLpcLTU1NitaKDs/4UdJfPB5HQ0MD\nHA4HqqurMTg4iHv37mlUYXHu37+PoaEhAMDQ0BDu3r1bcK0e5qdkFr/27Pf78fXrV+RyuXKU+9uU\nPmDXRhIAAAJWSURBVNf0MKvldHV1YcuWLQW/rufZAav3B/z+7Nb0vXpMJhN6enrQ0dGBa9eulbuc\nkkqn06ivr198bLfbkU6ny1iRcrlcDlarFQBgtVoL/hLpZX5KZrHcmlQqpVmNxVDSn8lkwvPnz+H1\netHb24uZmRmty/xj9Dw7JdTMTvWVu6vR6hqAcim2v7V+0Vqh/i5evPiPxyaTqWAva3l+v1I6i3+/\nqlrrM/w/JXW2t7cjmUxi06ZNiEajGBgYwOvXrzWoTht6nZ0Samb3x4K/0q8BKLa/uro6JJPJxcfJ\nZBJ2u73Yskpmpf6sViuy2Sx27NiBDx8+wGKxLLtuLc/vV0pm8e81qVQKdXV1mtVYDCX91dTULH4c\nDAYxOjqKz58/V8Q9tvQ8OyXUzK7sh3oKHZuqlGsACvXX0dGBN2/e4P3795ifn8fNmzcRCoU0rk6d\nUCiEiYkJAMDExAQGBgaWrNHT/JTMIhQK4caNGwCAWCyGzZs3Lx7uWuuU9JfL5Rafq/F4HCJSEaEP\n6Ht2Sqiandp3motx584dsdvtsmHDBrFarXLkyBEREUmn09Lb2ysiIm/fvhWv1yter1fcbreMjY2V\no1RVlPQnIvLw4UNpamoSp9Opq/7y+bwcOnRoyemcep7fcrMIh8MSDocX15w6dUqcTqe0traueDba\nWrRaf1evXhW32y1er1cOHDggL168KGe5v2VwcFBsNptUV1eL3W6X69evV9TsVutPzeyKvkkbERHp\nS9kP9RARkbYY/EREBsPgJyIyGAY/EZHBMPiJiAyGwU9EZDD/A3p8jeZ31YxVAAAAAElFTkSuQmCC\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x864d510>"
+       ]
+      }
+     ],
+     "prompt_number": 125
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "scatter([arctan2(p[0], p[1]) for p in c], [p[0]**2. + p[1]**2. for p in c])\n",
+      "title(\"$r$ vs Angle\")\n",
+      "xlabel(\"Angle (rad)\")\n",
+      "ylabel(\"$r$\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 126,
+       "text": [
+        "<matplotlib.text.Text at 0x8ea09d0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
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x0hJSaGrawsCO7S4mhBw5NKl71S6j3q3bwjVaYtuWvlLGRPTUVeZ58+f124+bKz9DSKQT0VxqNG0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\nAORP/uRP/uSvmn96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+       "text": [
+        "<matplotlib.figure.Figure at 0x8cab9d0>"
+       ]
+      }
+     ],
+     "prompt_number": 126
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Perfect circle $\\implies r = 1 \\forall \\theta$"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
     }
    ],
    "metadata": {}

UCC git Repository :: git.ucc.asn.au