{"nbformat": 3, "worksheets": [{"cells": [{"source": ["# Orthogonal Projection"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["import numpy as np\n", "import numpy.linalg as la"], "metadata": {}, "cell_type": "code", "prompt_number": 1, "collapsed": false, "outputs": []}, {"language": "python", "input": ["\n", "# for in-line plots\n", "%matplotlib inline\n", "\n", "# for plots in a window\n", "# %matplotlib qt\n", "\n", "import matplotlib.pyplot as pt\n", "from mpl_toolkits.mplot3d import Axes3D"], "metadata": {}, "cell_type": "code", "prompt_number": 2, "collapsed": false, "outputs": []}, {"source": ["Make two random 3D vectors:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["np.random.seed(13)\n", "x = np.random.randn(3)\n", "y = np.random.randn(3)"], "metadata": {}, "cell_type": "code", "prompt_number": 3, "collapsed": false, "outputs": []}, {"source": ["Make them orthonormal:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["y = y - y.dot(x)/x.dot(x)*x\n", "x = x/la.norm(x)\n", "y = y/la.norm(y)"], "metadata": {}, "cell_type": "code", "prompt_number": 4, "collapsed": false, "outputs": []}, {"source": ["Check:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["print(y.dot(x))\n", "print(la.norm(x))\n", "print(la.norm(y))"], "metadata": {}, "cell_type": "code", "prompt_number": 5, "collapsed": false, "outputs": [{"stream": "stdout", "text": ["7.6327832943e-17\n", "1.0\n", "1.0\n"], "output_type": "stream"}]}, {"source": ["Plot the two vectors:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["fig = pt.figure()\n", "ax = fig.add_subplot(111, projection='3d')\n", "ax.set_xlim3d([-3, 3])\n", "ax.set_ylim3d([-3, 3])\n", "ax.set_zlim3d([-3, 3])\n", "\n", "xy = np.array([x, y]).T\n", "ax.quiver(\n", " 0, 0, 0,\n", " xy[0], xy[1], xy[2],)"], "metadata": {}, "cell_type": "code", "prompt_number": 6, "collapsed": false, "outputs": [{"prompt_number": 6, "output_type": "pyout", "text": [""], "metadata": {}}, {"png": 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iq5qEvpKQJODLLwV8+GEI779fgw8+iOPddwP47jsBbZc9gWnTcli8OIPa2vId\nrFiLWMtd5Oz5WO9V4DR6syfIt09pgk4GRNlnzG+7RgA+F127LF1aUuVyOYTDYdPWnZV5sEEOs1vk\nmHWz0ENDfSqM5Pm6ZeEsWxbGL38ZRTwu4ZhjJDQ3S/j66yCCQWDEiDweeSSNCy9M4I9/DOG667Ko\nq0PJDlasa0ItYFeNecB2onRP0BZK8XjclZ076LNNTU2+s3Qr+zWrE7MPPmUDNDY2Ip/PWxJcs/Og\n9K8jR45AkiREo1E5Fcvp8Smy3dTUBABIJBKIxWKe5DyXIpMBOneWMG6ciJYWAQ0NBcRiQJ8+BTz9\ndCveeKMVp5wi4bHH0sjlBDz+uLqFThZbKBRCNBpFPB5HTU1NUfUe22c4m822a1LjN9xM4yIxDYfD\n8vVNJBJF15fEOZVKoaWlBel02vD1VboXOnfu7OSp2Y6vLV3CamFDIpGQI+RuzUPNd2olz9YIbJAs\nHA6joaEBqVSqonyYhQLwl78E8cwzIbzwQhj9++fxk5+I+N73JPzpTyHccEMzrrwyANb7c9ZZeQwb\nlscf/hDCqafmceGF+r5PtkcrQY3uSSh4wM4cet0TdH2Vfng19wQruk1NTTj55JNdPSer+Fp0zbgX\ntBrjUHDAKnqOodWQhnYrMIuePF+ngmT2XDvggw8CePrpMJ57LoQuXSRccomId95JoVevtuNv3BjE\nwoWtiERaEQy2z8986KE0vvxSwBVXxDFgQB59+5qbFwkFm2pVKs1Kbflc/nydt0K9KFjQg9HiDq0G\n5tyn6xF6CiTKFTbYFYzTM4d8Pq/aCc2uQg81lEEyZeqZ2fO344H+/HMBzzwTxrPPhpBOC7jkkhxW\nr25F//7tr8Wpp+ZRKABaG1306SOhTx8JN9+cxWWXxbFuXQtM5M6roiUU5bYnLxew47ShN18baGtg\nPmXKFHTq1El2xw0ZMgR1dXW6x3vmmWewaNEifPLJJ3j33XcxYsQI289JDV+LLt3AgUBAjp4qMVrY\nYHU+asdg832dnIPa59nzNxokc5JvvhHw/PMhPP10GHv2CLj4YhG/+lUaY8YUYMf0pk/PYdOmIK69\nNobly9O2HFMNVogJvQE7+r1qwCmLWinE9Cx17twZd999N5YuXYo9e/bg+uuvx3fffYddu3bpPvbg\nwYOxevVqXH311bbPuxS+Fl2iEgob1I5RboscJ2FzjaPRKDp16lQ2z9dpAWhqAv70pzahfe+9ICZN\nEjF/fgbvUEYaAAAgAElEQVRnnZWH3X2CBAFYujSNH/84gccfD+PKK61tPGps7KNCwbpvWIuYXBMU\nqFNLs7IDN90LbhIOh3HaaadhyZIlePTRR1FXV2d4DqeccopDsytN1YkuZQOwzWD0CJ2dub7KQJWR\nRuZWLV1lrrFbQk+RfhIMug7ZrIC1a0N45pkQ1q0L4f/9vzymTcth1apWJBLOzimRAH7721ZMmJDA\n8OF5jBql33XjhFgpfb0tLS1FPSz8HrDzwnfc2tqKxN9vpEq/PoSvRZcNpFVCYQPQ5js128jcap4v\npYCZbaJuBvKxpdNpeTw2Uf6OO+qxbVsIU6Zkcd99rTjmGHcfjj59JCxfnoYBV5+raLkn7ArYuZ0y\n5sY4yvMpNe6ECRNw4MCBdv9+55134oILLrB9fnrwtegSFLzI5XKWxcbsTZrL5ZBOpyFJ1hqZm4GC\nZNTDNpFIGD4Hszm+uVzbsr2mpkbOAAkGg3JD9dtuywJIM43NpXZLaaetuB//WN3fX6nojex3xIAd\n+3zquV/Xrl3r9JQM42vRlaS27WnoizASuVRCN6pR0WVT0Kgyx62dI5RBMjfaH7LuG/JTNzY2Ip1O\ny+OSEJMoAALC4XBR8Eitjt9NIfYb5SL7WgE7spCdtni97FVhx7hu+qR9LboktIVCQa6qsno8vRdf\nLSuAqpmcHl8rSGalibqRHF9yX5AvMh6PQxSPNpoBIJfSslYXm2EiCAJCoVDRC0rLr+lEgKkSsCpU\negJ2JMKiKFbF9WSvmSiKhlyISlavXo05c+bg4MGDOO+88zB8+HC8+uqrdk1VE1+LLgD5otNb3+pN\nXE70lBkJbFaA0xkA5QJ0To2vzPEl9wGNRctdKv9kA0P0p1AoqD7sbF4yCTH7HaoJMZ1nPp/3pXA4\njVoRQTgcLmpOY3fAzosiDKsdxi6++GJcfPHFdk1NN74XXcCdRuZ6siKcyLOlsZ1ut6hWmMH6bamY\ng6wnAHIdfSAQQE1NTdELIBQKFc2RRFIpxOzDzo7LzkspxKIoIp/Py5ti+jHS7xbkuikXsKP0Na1G\n5pXYDc2PzW6AKhBdtkDCieIGaoqjtkWOns9bHZ/1GZfbVt0uS1fNb0uiSWOk02k5WKYURa25GRVi\n1s+udl60dTkA2ze/7AiUKuwwGrDzIkvCj710gSoQXcLu4gayLmmLHLZHgtPQjc9amW5UkrHuC6Xf\nlv15LpdDNBot21i9HHqEmESUreIitwJrEUuS1E781XrpVpIQexl80sJowI5thUnX2K2AnR/7LgBV\nILp2+lPpGEasS7XPm32Y6DNskMxIcYOai0AvrFWj5rfN5XLIZDIIh8Oora11bLmpJcRUfAFAFly1\nPdKU5x8MBuVj0QuEPksZF2pL6UoURDPYcR5aATvWIgaATCYjC7GTFXYEdy94jB2iy1p6ZqxLKzcW\njQ20+UqNFneYhfXbkr+YLERBaOsvq+W3dQMKXIqiiFgsJr8AWcuLLGKygMsJMSseSiFmXSjU9a3a\nc1/NQi9IeilSZVi5gJ3VVQZ97siRI1x0vcSK6LJ9GkKhUNk+BXrmoffzSjcGAFeamCvTzuLxODKZ\nDLLZrLxcbG1tNeS3tRPWuo1EIrKrgygVGConxGp+YqUQU+qfWntBO4oQ3MoLddNip3HKBeyUWycZ\n2dqHPZ/m5mb07t3b2ZNyAN+LrhX3gjIjIRaLuZJ2Rqi5MRobGx1PO6PAYCgUksWM8jgpQAa0iYxV\nv60ZaOlv1Lo2IsSsf1cpxKyLiPz45Lopt82PUSGuBuu53DPjVMCOW7oeY9TKU8t3pWWsVfTk+moF\nyaxY7OU+q2yeTr5RNiAiiiLC4TDC4bD8UNADQb/D/rFTNCgFjaxrOwKXRoWYxJX17bLXVBAE+UXE\nBphEUVStrjMjxB2BcgE75R5rbNoa/by5uZkH0ryAFatyQaRy+a52PBjllkZ6gmR2i66yeo7Nt2X9\ntpSloWZZskJFAsOmd1kRYnoJZrNZRCIRU70jjKAmxGw1Ib1waPskVjyVWROEmhCXKnN2S4TdcC/Y\nNYaaENPx2R2Hv/nmGwwbNgw9e/bEt99+i7POOgsjR47E2LFjdY81b948vPTSS4hEIujduzeefPJJ\n16xmocwDXvEdllnfXy6XQ21trervkd9UkiTE43HVjAQ6jpUeDk1NTfLx2TmylnU8HtdcMqt9Xi90\njvX19fK4rMiT+4TEloJUZv225fJsywkxvQSpQ1ksFnM9CZ9cTNTknr0v2CUw+wcoFmKt4yqtZbZz\nGD13oVDIUTFOpVJyXw6noEKVhMO9Oin2UlNTg+bmZlx55ZU4/fTT8dlnn6GpqQlPP/207mOtXbsW\n48ePRyAQwE033QQAuOuuu+ycruYX6XtLlyhl5ZXaIkfPMazMg/yngiDoqiSzw71AL6KWlhbZfQIU\n9z4gobGSb6snz1bLIqY5SFJbabEbbShZlIKvlgbHWl5q3b6UQsz6INmcYvZ4oVAIkUhEFqpAIOBo\nvwk/WbpGqKurQyqVwnXXXVdUJKOXCRMmyP8/duxYPPfcc3ZOryS+F10tX6jeLXKUx7Iz15e28jaT\n62sWSZLk5j8kJOxymPJttYTGKnqEmPI56XdZ0XLj4SV3ixnBVxNioDhNiv4oA23svUHfCV0DtqJS\nGbDjZc5tKMWd4g9W+fWvf42pU6daPo5efC+6QHFbRr1+U63j2JE5QEv2eDyuS+ztgISkUCjI1XOs\n35Z+Xspv6xTkQ6WAUzgcRiQSKRJjO33EWrCuBDsq6lhIFPUIMcG+iJV7/KlV12mVOWsJcbWlpbHj\n0LmVGldPA/M77rgDkUgEl156qQMzVqcqRJcoFAo4cuSIqZ0jAOs7N6TTaTnX1+w2OUbnwL5kSETI\neqRjkcUdj8ddz7cFjgap1ATfjGvCqBDrcSU4ASvE5PJJp9Nyb2E2Z7VcBzZAXYiVjWrUmtRUi3tB\n7bkoNW65BuYrVqzAK6+8gnXr1lmemxGqQnSz2SxSqRQkSUJ9fb1p/yBrLeu9iZT+00gk0q5rlpk5\nGB23vr5eDtg1NzcX+RTdyAhQQ6uaTAsrPmItIWbT0LzwHQNHXzqBQAC1tbXtjAGtgKQeIdYqc6br\nBEDupaEUYz+itHTNsmbNGixZsgTr169HLBazY2q6qQrRJb9pMpm0tGw2KkqUfsYGychX6CTsuEq/\nbW1trey3pQcsm83KDZ+dyrFlKVdNZgQzQkw+0lwu54grQS+lMiNY9JyjlhAr84jJj0zluWTdqxUg\nGKkEK3eebrsX0uk04vG46WPNnj0b2WxWDqiddtppePjhh22ZZzmqQnQTiYTpRi9K6EYudROVCpLp\nyRfWM74aykyMUn5b1qLSEimlCNsRoDFbTWaEciKVzWaLUrtIcNwKQrHujHA4bOqlY0SI1fy69HO6\nn5QWMRus01MJVgnYWY22a9cuu6ZlmKoQXaXgWbV2tUSPzYjQCpJZDcapiTYt0zOZDGKxGGpqauQH\nkMZnxVjp+yv3AIuiKPsFzQqxE9VkRiCBEEURkiTJgm/nOerBSXeGESEmWIOgnGuiVJlzKSGmnGQ3\n8WuHMaBKRJdg027MoiaatExjm3pr3WR2ZUDQuMpyZUA939ao39YuIXa7mkwNdg5KV4LTLxs9c3AS\n9nukOZBbh3KAzXRgY3+uLMlV9ptwM0uCnrsjR474sgQYqBLR1crVNXssOoZasMrpVCsav5TfFoBc\numtnNF5NiOmhE0WxaJt58gmLoohQKORJ20eg2J2h5zo4YfUrA2VeBKrYOdTV1amW0hrpwEYCS/9G\nLxE2WMf2m6D7RE3Y7YJ1L3BLt0KwU3TVgmRuzIHGFkWxrN/WjWg8PUDsOCR0JEz5fB6pVEo1UOeU\ntUeuHjvcGWaFWBAE2X/sVTqekWCd3sY/rBVLn6XfZY9HQpxOp+XrUarfhB3VdYRft+oBuOi2g/Ja\nqVpJbyWZ1TmQ35asNkoBU/Pb6km/cgLKCVXLCGAfXjaQpWYtWsEtd0YpISYLjz1H8iW7WTFGvTbM\nrnb0CDEbkFP6ddnqOjoGW/gCqO/mbKXMmX63sbERXbp0MXS+lUJViK4d7gWynKhiymwlmdE5sH5b\nEhEqsqAbshJ8ppQCprVdj7IiS9mjgNLW2AfdqBAbdSXYDRusA46WWbsdrGPzn802R9LCiEVMUIqe\nmkUMHA3mlRPicmXOrHuhubkZJ554om3n7SZVIbqEGdFlg2SRSATRaNTSA2JkDmp+W2omzub7BgIB\nRKNR1zMCgNLVZKUggVIKMfvwZjIZeTlbKoeYXohsqpzbVEqwzmoqmhmUQkzN9+m7Jf8u2/yevkMK\nbqtV16n1m1BuRMoKMSu6jY2N6Ny5s+Pn7gRVIbpmLF2y3pRbq7tR3FAq3zYYDCIajcoBKxIYengB\n+5fsahitJtODHitK2TCdlrBeWfnAUQs7GAx6FqxjfdheVdaxLx61e0K5uqE/QPkObIC6ELNlzkCb\ni2358uU4dOhQReUQG6EqRJdQy3FVQ7mDAms56T1GqTlolRIrO5+p5duSi0NL6JS+U+WSnfqzmr0h\n7awm04OWEJPQAfCkqg6wdxlvVohp9eOlewnQl6GhtbopJ8TssZRCrKyuy+Vy2Lt3LzZs2IBnn30W\n3bp1w4QJE/DYY4/pPpeFCxfixRdfhCAI6NKlC1asWIFevXqZuSym8H0Tc+CoUJRrQs5amFpBsnLN\n0PXw3XffoXPnzu1yWclvS7XerLjT/MnFoffB0gp86O1NwML6TGOxmCcpYEqho4wA9jzJp6gs/bXL\nd6p88RjdFdrq2KwQk/+YMkic8BGXmw+JnV2uHb0WMfv80PWg7+InP/kJVq1ahUOHDmHfvn04++yz\ndY/f3Nwsa8SyZcuwfft2PP7445bPS0F1NzEv514w0lvXzrQzSilitwdSWtJW8231LNnVehOwN7fX\n1WQ0Z727/0YiEfkzdvtOKSUPgCe5x3Se9GKhFw/7wnEjWAcctW71ulX0omYRA9qtMOl52rJlC7p1\n64YPPvgAH3/8MaLRKPr164d+/foZGp81ypLJJLp27WrLeemlKkQXKO6pSyiDZHraLdoluvRwkFVN\nDw6bb0vLZyfKRdWEWE2g6HxDoVDJbYScxEqwzi7fqd58V6fRWsYrc6WdCtbRse22bvWgzIARRRGp\nVEo+99WrV+O1117Dt99+i9GjR+OWW27BLbfcYiqgtmDBAjz11FNIJBLYuHGj3adSkqpwLwCQfX7N\nzc1oaGgoCpIlEgndDzJ90WYTrwuFAhobGwFA3h6ILE8SOLsDVEYhnyldH3ohsLmmSv+wE3N0Ilin\nhlZ/AvbFRH2Qnd5PrNQcyZ9vVujKnaceIabMBLoWXrx4lKIfCoXw8ssvY8mSJbjjjjswcuRIbNu2\nDVu3bsXcuXNVt+vR08AcaNsXbefOnXjyySftPg3NC1c1oktVXEeOHJEfJPLbGiGfz5va2pn12wJH\nl6Z2+G3thJbPkiTJNzSL2hIPsDdjQpn3G4vFPCn00LL4lX5wJ+dG2RnUX8Pua6EUYuWLlT1Pp3J/\njcC6NOLxOJqamvCLX/wCgUAA999/v+1pYl9++SXOPfdcfPTRR7YeF9Xu0wWOWqgAEIvFLG22aLS4\ngbIhgsGgvGEeCQolz7u9a4GSUtVkLOWKHMhlQi4MVqT0XG+zrgQ7oe9MGShzq6qOcCMNTM0Fo9VP\nQxAEeZcLCsa6GUBUWrdvvvkmFi1ahP/4j//ARRddZNtcdu3ahb59+wIAXnjhBQwfPtyW4+qlakSX\nRI06brlR3EBLMXpowuEw8vm2nRpIXOhYbCcotXQyp9BTTVYKPUUOytxatYwJt1wJ5Sgl+nqq6uwQ\nYmXQ0O00MLbwgI0rAJC/T7axkZPBOqC9H7u1tRU33ngjDh06hFdeeQXHHnusrePNnz8fO3fuRDAY\nRO/evfHII4/YevxyVI17gaK9hw8fNr0/GdD2QBw+fLgo5UuJsq9uKb8tVbix0We1ZawT1i8rME4H\nydRKRSljggKHXvtM7QiUKYWY7ju9udJsdoRXgUvgaN+GUi4NrWwCu4RYWWwRCoWwadMmzJ8/H9de\ney0uvfRS3xZAoCO4Fwir2QdsFoTyC1dmQ6j1t2X7JGgVFpRaxhpdrqvhhVWpljFBol8oFORgXXNz\nsyvWE8H6TO1w7+ix/NPpdDvLn4ocvNw+CDh6b7BZNVooLX/6PLvCMWsRU848WbfZbBa33norPv30\nU6xevRo9e/a09bwriaqxdMnqaGxsNBVAYzl8+HBR71yl35aiulr5trFYzPBy044CBy+T+pXz0LIq\n1axEemiVLxyrc2d9pmpBQydhv1MK8gIw/J3aOR+2b4OdATsjFjGAdqXE27dvxw033IDp06fjiiuu\n8GQl5AAdx9Jla7fNwlrL5LeVJKnIb6vMt6Wfm3mwyxU4UBloqfQfN/YmK4fywVazKtnADqX62B3A\nYpetXpXOkjWs7LfLvnT0+MLtwOmAnRGLGGi7Nn/+85/Rr18/rF69Gps2bcLvfvc7nHTSSbbOq1Kp\nGkuXthRJJpMIh8OquXt6aWxsRCwWk8WuUvJttdJ/aD5slzS3YX2V5J8zSym/qVpKFwsblPHKfwwU\n97ottfKhe4rOkS1vtlr2y74EvV75ZLNZ+WWcz+cxbdo0bNu2DclkEqeeeirGjh2LO++809L80uk0\nzjzzTPmFe+GFF2Lx4sU2nokhOo6la9WnSw98KpVCLBYz7bd1AtZKJGsuk8nID2WhUEAymQRgn3+4\nHKwrwS5fpZmMCfKZsr7KSveZAsaqB434TVnr1quVD82DmktRP5OHHnoI6XQa69evR5cuXbB161Z8\n/vnnlr+vWCyGN954A4lEAqIoYty4cXjnnXcwbtw4y+dhJ1x0/w7rt6XCgUgkouq3DQaDFbGED4VC\n7fbDMpPOZcc8nM4/LiVO5Mem86E5uRGoY+eidK1YSVs0W95MJehk3XrVlYy9HvQy/r//+z/MmTMH\nZ599Nl5//XX5hTp58mTbxqXUN3LrHHPMMbYd2y6qRnTpxjIjukq/rbJlIpVoWvHb2gFbTaY1Dzv8\nw3bMww3IqpSkti3XlT5TN7ddp+vh1MtYjxCzflMqzGH9qG5BVjZdD0EQ8MQTT+APf/gDHnroIUeL\nEQqFAkaMGIHdu3dj1qxZGDBggGNjmaVqRJdQZhWUgpY+VAVDfttgMCj75NjiBtrW2m30VpNpwQqx\nVocuemBL5Q874UowAxsoU86jlDgpK7Cslvyy2SJeXA/Wx02GQiQSQSgUKgpgAe2Dkk6UN6v5kPfv\n3485c+Zg2LBheOONNyzFWvQQCATw/vvvo7GxERMnTsSbb76Js846y9ExjVI1omvE0mXzbaPRKBoa\nGuQHE0BRK71wOCz/nYRYT0DHDqxWk5VCqzxUK4uAMiS8LGUGzG13bvRc9WRMUJ6pl+XMNA+1VpRa\nmQROlTcrfciCIGDVqlVYvnw5li5dih/+8IcWztI4DQ0NOO+887BlyxYuuk5TytIlEaOGGhQEY4Nk\ndBNr+W1L+Uzt9CN60aNArQyWrGCKqFOPC7dzTe3ejLFUyS/bllP5gg0EAvKqw8tyZiNWttPlzWSM\nkA/522+/xfXXX4/jjz9eDmy5wcGDBxEKhdCpUye0trZi7dq1uPXWW10Z2whVJboU8VazdJV+W2V/\nW9YPVcpPaTTarHxgy1EpPQq0lvB2+4f1zMONzRj1ZEzQuQIo8h+7VeBAaFm3etE6V62XjlYmjDJT\nIxgM4sUXX8R9992Hu+66C2effbar1+Wrr77C5ZdfLj/Xl112GcaPH+/a+HqpmjxdoLinLrVmZP22\nVNxQqk+CXX45rRaJWm6JSqomY8tm9VTXaeUPW+0vUSl9CpQvwmAw2C6v1slAHeG2D7lUpSStEBsb\nG9HQ0IBCoYB58+YhFoth6dKlpvtRVxEdI0+X7ZtQzm8LoMhfarcFpbak03JLUPvHQCDgeTaAmcol\noz7TcvnDpQJlblLKyrY7r7Yc7AvILZ+62qqO7hFRFBEKhfDss8/izjvvRCwWw7Bhw3DhhRfim2++\nsSy6e/fuxbRp0/DNN99AEARcddVVmDNnjtVTqgiqytJlG5mTj5VqzFk/L5XuerkBI4B2/lISZjNu\nCSu4IXJ6+0uQVWmmh4WdWO3bwC7V2ZWOUZ+pshOXV+4moP2uEslkEgsWLEAqlcKVV16Jzz//HFu2\nbMGkSZNw0UUXWRrrwIEDOHDgAIYNG4ZkMomRI0fij3/8I/r372/T2ThO9e8cAbRtX97c3Ix8Po/a\n2lrZb0ulsqzf1qsNGIH2vQFYV0KpnRucqDBjeza4XTar9A/ncjkA3u18S3Nyys1jdFcOypDwuqSZ\nTRWkF9Bf/vIX3Hzzzbj++uvx05/+1PHv56KLLsLs2bMr0kerQcdwL5DPLZVKyT4ntkqpEvJLy1Vx\nGXFLWBEm1pLz6gVEwRzaapy+G1aI9eQ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C0WjUlY0yWdS2z9m4cSPmz5+Pa6+9Fpdeeqmn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\n", "text": [""], "output_type": "display_data", "metadata": {}}]}, {"source": ["Make an array with the cornerpoints of a cube:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["points = np.array([\n", " [-1,-1,-1],\n", " [-1,-1,1],\n", " [-1,1,-1],\n", " [-1,1,1],\n", "\n", " [1,-1,-1],\n", " [1,-1,1],\n", " [1,1,-1],\n", " [1,1,1],\n", "])"], "metadata": {}, "cell_type": "code", "prompt_number": 7, "collapsed": false, "outputs": []}, {"source": ["Plot them:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["fig = pt.figure()\n", "ax = fig.add_subplot(111, projection='3d')\n", "ax.scatter(points[:,0], points[:, 1], points[:, 2])"], "metadata": {}, "cell_type": "code", "prompt_number": 8, "collapsed": false, "outputs": [{"prompt_number": 8, "output_type": "pyout", "text": [""], "metadata": {}}, {"png": 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OvhctN0Y70rUYRLZshxhLtlKapNmVB3LrJLIlGUEv+ZM8oRY8zwvddYlEQrfMkms6qNZO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\nGxbC1nRt2LBhw0LYpGvDhg0bFuL/Azr85codNAjmAAAAAElFTkSuQmCC\n", "text": [""], "output_type": "display_data", "metadata": {}}]}, {"source": ["Construct the projection matrix:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["Q = np.array([\n", " x,y,np.zeros(3)\n", "]).T\n", "print(Q)"], "metadata": {}, "cell_type": "code", "prompt_number": 9, "collapsed": false, "outputs": [{"stream": "stdout", "text": ["[[-0.68624693 0.65133881 0. ]\n", " [ 0.72610421 0.63968157 0. ]\n", " [-0.04286988 0.40812404 0. ]]\n"], "output_type": "stream"}]}, {"language": "python", "input": ["P = Q.dot(Q.T)\n", "print(P)"], "metadata": {}, "cell_type": "code", "prompt_number": 10, "collapsed": false, "outputs": [{"stream": "stdout", "text": ["[[ 0.89517709 -0.08163735 0.29524635]\n", " [-0.08163735 0.93641985 0.22994143]\n", " [ 0.29524635 0.22994143 0.16840306]]\n"], "output_type": "stream"}]}, {"source": ["Check that $P^2=P$:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["la.norm(P.dot(P)-P)"], "metadata": {}, "cell_type": "code", "prompt_number": 11, "collapsed": false, "outputs": [{"prompt_number": 11, "output_type": "pyout", "text": ["1.7880278822442372e-16"], "metadata": {}}]}, {"source": ["Project the points, assign to `proj_points`:"], "cell_type": "markdown", "metadata": {}}, {"language": "python", "input": ["proj_points = np.einsum(\"ij,nj->ni\", P, points)"], "metadata": {}, "cell_type": "code", "prompt_number": 12, "collapsed": false, "outputs": []}, {"language": "python", "input": ["fig = pt.figure()\n", "ax = fig.add_subplot(111, projection='3d')\n", "ax.scatter(points[:,0], points[:, 1], points[:, 2])\n", "ax.scatter(proj_points[:,0], proj_points[:, 1], proj_points[:, 2], color=\"red\")\n", "\n", "xy = np.array([x, y]).T\n", "ax.quiver(\n", " 0, 0, 0,\n", " xy[0], xy[1], xy[2],)"], "metadata": {}, "cell_type": "code", "prompt_number": 13, "collapsed": false, "outputs": [{"prompt_number": 13, "output_type": "pyout", "text": [""], "metadata": {}}, {"png": 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F3qIdhFFkC2XLtQOxqhgQe8858oLNMBpjStlbHMcZsmyNzoNibePxuFjXVuv4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\n3NTGAJs9rlwSB4Xn8TyPuro69O7dG2+++SaefPJJPPzwwzj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