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nicoguaro/bilinear_map.ipynb

Last active August 29, 2015 14:19
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Bilinear map inversion (analytically)
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bilinear_map.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from __future__ import division\n",
"from sympy import * \n",
"x, y, u, v = symbols('x y u v')\n",
"a0, b0, c0, d0 = symbols('a0 b0 c0 d0')\n",
"a1, b1, c1, d1 = symbols('a1 b1 c1 d1')\n",
"init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"eq1 = x - a0 + b0*u + c0*v + d0*u*v\n",
"eq2 = y - a1 + b1*u + c1*v + d1*u*v"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"sol_u, sol_v = solve([eq1, eq2], [u,v])"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
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],
"text/latex": [
"$$\\left ( \\frac{1}{2 b_{0} d_{1} - 2 b_{1} d_{0}} \\left(a_{0} d_{1} - a_{1} d_{0} - b_{0} c_{1} + b_{1} c_{0} + d_{0} y - d_{1} x - \\sqrt{a_{0}^{2} d_{1}^{2} - 2 a_{0} a_{1} d_{0} d_{1} + 2 a_{0} b_{0} c_{1} d_{1} + 2 a_{0} b_{1} c_{0} d_{1} - 4 a_{0} b_{1} c_{1} d_{0} + 2 a_{0} d_{0} d_{1} y - 2 a_{0} d_{1}^{2} x + a_{1}^{2} d_{0}^{2} - 4 a_{1} b_{0} c_{0} d_{1} + 2 a_{1} b_{0} c_{1} d_{0} + 2 a_{1} b_{1} c_{0} d_{0} - 2 a_{1} d_{0}^{2} y + 2 a_{1} d_{0} d_{1} x + b_{0}^{2} c_{1}^{2} - 2 b_{0} b_{1} c_{0} c_{1} + 4 b_{0} c_{0} d_{1} y - 2 b_{0} c_{1} d_{0} y - 2 b_{0} c_{1} d_{1} x + b_{1}^{2} c_{0}^{2} - 2 b_{1} c_{0} d_{0} y - 2 b_{1} c_{0} d_{1} x + 4 b_{1} c_{1} d_{0} x + d_{0}^{2} y^{2} - 2 d_{0} d_{1} x y + d_{1}^{2} x^{2}}\\right), \\quad \\frac{1}{2 c_{0} d_{1} - 2 c_{1} d_{0}} \\left(a_{0} d_{1} - a_{1} d_{0} + b_{0} c_{1} - b_{1} c_{0} + d_{0} y - d_{1} x + \\sqrt{a_{0}^{2} d_{1}^{2} - 2 a_{0} a_{1} d_{0} d_{1} + 2 a_{0} b_{0} c_{1} d_{1} + 2 a_{0} b_{1} c_{0} d_{1} - 4 a_{0} b_{1} c_{1} d_{0} + 2 a_{0} d_{0} d_{1} y - 2 a_{0} d_{1}^{2} x + a_{1}^{2} d_{0}^{2} - 4 a_{1} b_{0} c_{0} d_{1} + 2 a_{1} b_{0} c_{1} d_{0} + 2 a_{1} b_{1} c_{0} d_{0} - 2 a_{1} d_{0}^{2} y + 2 a_{1} d_{0} d_{1} x + b_{0}^{2} c_{1}^{2} - 2 b_{0} b_{1} c_{0} c_{1} + 4 b_{0} c_{0} d_{1} y - 2 b_{0} c_{1} d_{0} y - 2 b_{0} c_{1} d_{1} x + b_{1}^{2} c_{0}^{2} - 2 b_{1} c_{0} d_{0} y - 2 b_{1} c_{0} d_{1} x + 4 b_{1} c_{1} d_{0} x + d_{0}^{2} y^{2} - 2 d_{0} d_{1} x y + d_{1}^{2} x^{2}}\\right)\\right )$$"
],
"text/plain": [
"⎛ ____________________________\n",
"⎜ ╱ 2 2 \n",
"⎜a₀⋅d₁ - a₁⋅d₀ - b₀⋅c₁ + b₁⋅c₀ + d₀⋅y - d₁⋅x - ╲╱ a₀ ⋅d₁ - 2⋅a₀⋅a₁⋅d₀⋅d₁ + 2\n",
"⎜─────────────────────────────────────────────────────────────────────────────\n",
"⎝ \n",
"\n",
"______________________________________________________________________________\n",
" 2 2\n",
"⋅a₀⋅b₀⋅c₁⋅d₁ + 2⋅a₀⋅b₁⋅c₀⋅d₁ - 4⋅a₀⋅b₁⋅c₁⋅d₀ + 2⋅a₀⋅d₀⋅d₁⋅y - 2⋅a₀⋅d₁ ⋅x + a₁ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅d₀ - 4⋅a₁⋅b₀⋅c₀⋅d₁ + 2⋅a₁⋅b₀⋅c₁⋅d₀ + 2⋅a₁⋅b₁⋅c₀⋅d₀ - 2⋅a₁⋅d₀ ⋅y + 2⋅a₁⋅d₀⋅d₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2⋅(b₀⋅d₁ - b₁⋅d₀) \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅x + b₀ ⋅c₁ - 2⋅b₀⋅b₁⋅c₀⋅c₁ + 4⋅b₀⋅c₀⋅d₁⋅y - 2⋅b₀⋅c₁⋅d₀⋅y - 2⋅b₀⋅c₁⋅d₁⋅x + b₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
"2 2 2 2 \n",
" ⋅c₀ - 2⋅b₁⋅c₀⋅d₀⋅y - 2⋅b₁⋅c₀⋅d₁⋅x + 4⋅b₁⋅c₁⋅d₀⋅x + d₀ ⋅y - 2⋅d₀⋅d₁⋅x⋅y + d₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"_____ ______________________\n",
"2 2 ╱ 2 2 \n",
" ⋅x a₀⋅d₁ - a₁⋅d₀ + b₀⋅c₁ - b₁⋅c₀ + d₀⋅y - d₁⋅x + ╲╱ a₀ ⋅d₁ - 2⋅a₀⋅a₁⋅d₀⋅\n",
"─────, ───────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 \n",
"d₁ + 2⋅a₀⋅b₀⋅c₁⋅d₁ + 2⋅a₀⋅b₁⋅c₀⋅d₁ - 4⋅a₀⋅b₁⋅c₁⋅d₀ + 2⋅a₀⋅d₀⋅d₁⋅y - 2⋅a₀⋅d₁ ⋅x\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 2 \n",
" + a₁ ⋅d₀ - 4⋅a₁⋅b₀⋅c₀⋅d₁ + 2⋅a₁⋅b₀⋅c₁⋅d₀ + 2⋅a₁⋅b₁⋅c₀⋅d₀ - 2⋅a₁⋅d₀ ⋅y + 2⋅a₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2⋅(c₀⋅d₁ - c₁⋅d₀) \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅d₀⋅d₁⋅x + b₀ ⋅c₁ - 2⋅b₀⋅b₁⋅c₀⋅c₁ + 4⋅b₀⋅c₀⋅d₁⋅y - 2⋅b₀⋅c₁⋅d₀⋅y - 2⋅b₀⋅c₁⋅d₁⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 2 2 \n",
"x + b₁ ⋅c₀ - 2⋅b₁⋅c₀⋅d₀⋅y - 2⋅b₁⋅c₀⋅d₁⋅x + 4⋅b₁⋅c₁⋅d₀⋅x + d₀ ⋅y - 2⋅d₀⋅d₁⋅x⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"___________⎞\n",
" 2 2 ⎟\n",
"y + d₁ ⋅x ⎟\n",
"───────────⎟\n",
" ⎠"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol_u"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"YII=\n"
],
"text/latex": [
"$$\\left ( \\frac{1}{2 b_{0} d_{1} - 2 b_{1} d_{0}} \\left(a_{0} d_{1} - a_{1} d_{0} - b_{0} c_{1} + b_{1} c_{0} + d_{0} y - d_{1} x + \\sqrt{a_{0}^{2} d_{1}^{2} - 2 a_{0} a_{1} d_{0} d_{1} + 2 a_{0} b_{0} c_{1} d_{1} + 2 a_{0} b_{1} c_{0} d_{1} - 4 a_{0} b_{1} c_{1} d_{0} + 2 a_{0} d_{0} d_{1} y - 2 a_{0} d_{1}^{2} x + a_{1}^{2} d_{0}^{2} - 4 a_{1} b_{0} c_{0} d_{1} + 2 a_{1} b_{0} c_{1} d_{0} + 2 a_{1} b_{1} c_{0} d_{0} - 2 a_{1} d_{0}^{2} y + 2 a_{1} d_{0} d_{1} x + b_{0}^{2} c_{1}^{2} - 2 b_{0} b_{1} c_{0} c_{1} + 4 b_{0} c_{0} d_{1} y - 2 b_{0} c_{1} d_{0} y - 2 b_{0} c_{1} d_{1} x + b_{1}^{2} c_{0}^{2} - 2 b_{1} c_{0} d_{0} y - 2 b_{1} c_{0} d_{1} x + 4 b_{1} c_{1} d_{0} x + d_{0}^{2} y^{2} - 2 d_{0} d_{1} x y + d_{1}^{2} x^{2}}\\right), \\quad \\frac{1}{2 c_{0} d_{1} - 2 c_{1} d_{0}} \\left(a_{0} d_{1} - a_{1} d_{0} + b_{0} c_{1} - b_{1} c_{0} + d_{0} y - d_{1} x - \\sqrt{a_{0}^{2} d_{1}^{2} - 2 a_{0} a_{1} d_{0} d_{1} + 2 a_{0} b_{0} c_{1} d_{1} + 2 a_{0} b_{1} c_{0} d_{1} - 4 a_{0} b_{1} c_{1} d_{0} + 2 a_{0} d_{0} d_{1} y - 2 a_{0} d_{1}^{2} x + a_{1}^{2} d_{0}^{2} - 4 a_{1} b_{0} c_{0} d_{1} + 2 a_{1} b_{0} c_{1} d_{0} + 2 a_{1} b_{1} c_{0} d_{0} - 2 a_{1} d_{0}^{2} y + 2 a_{1} d_{0} d_{1} x + b_{0}^{2} c_{1}^{2} - 2 b_{0} b_{1} c_{0} c_{1} + 4 b_{0} c_{0} d_{1} y - 2 b_{0} c_{1} d_{0} y - 2 b_{0} c_{1} d_{1} x + b_{1}^{2} c_{0}^{2} - 2 b_{1} c_{0} d_{0} y - 2 b_{1} c_{0} d_{1} x + 4 b_{1} c_{1} d_{0} x + d_{0}^{2} y^{2} - 2 d_{0} d_{1} x y + d_{1}^{2} x^{2}}\\right)\\right )$$"
],
"text/plain": [
"⎛ ____________________________\n",
"⎜ ╱ 2 2 \n",
"⎜a₀⋅d₁ - a₁⋅d₀ - b₀⋅c₁ + b₁⋅c₀ + d₀⋅y - d₁⋅x + ╲╱ a₀ ⋅d₁ - 2⋅a₀⋅a₁⋅d₀⋅d₁ + 2\n",
"⎜─────────────────────────────────────────────────────────────────────────────\n",
"⎝ \n",
"\n",
"______________________________________________________________________________\n",
" 2 2\n",
"⋅a₀⋅b₀⋅c₁⋅d₁ + 2⋅a₀⋅b₁⋅c₀⋅d₁ - 4⋅a₀⋅b₁⋅c₁⋅d₀ + 2⋅a₀⋅d₀⋅d₁⋅y - 2⋅a₀⋅d₁ ⋅x + a₁ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅d₀ - 4⋅a₁⋅b₀⋅c₀⋅d₁ + 2⋅a₁⋅b₀⋅c₁⋅d₀ + 2⋅a₁⋅b₁⋅c₀⋅d₀ - 2⋅a₁⋅d₀ ⋅y + 2⋅a₁⋅d₀⋅d₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2⋅(b₀⋅d₁ - b₁⋅d₀) \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅x + b₀ ⋅c₁ - 2⋅b₀⋅b₁⋅c₀⋅c₁ + 4⋅b₀⋅c₀⋅d₁⋅y - 2⋅b₀⋅c₁⋅d₀⋅y - 2⋅b₀⋅c₁⋅d₁⋅x + b₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
"2 2 2 2 \n",
" ⋅c₀ - 2⋅b₁⋅c₀⋅d₀⋅y - 2⋅b₁⋅c₀⋅d₁⋅x + 4⋅b₁⋅c₁⋅d₀⋅x + d₀ ⋅y - 2⋅d₀⋅d₁⋅x⋅y + d₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"_____ ______________________\n",
"2 2 ╱ 2 2 \n",
" ⋅x a₀⋅d₁ - a₁⋅d₀ + b₀⋅c₁ - b₁⋅c₀ + d₀⋅y - d₁⋅x - ╲╱ a₀ ⋅d₁ - 2⋅a₀⋅a₁⋅d₀⋅\n",
"─────, ───────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 \n",
"d₁ + 2⋅a₀⋅b₀⋅c₁⋅d₁ + 2⋅a₀⋅b₁⋅c₀⋅d₁ - 4⋅a₀⋅b₁⋅c₁⋅d₀ + 2⋅a₀⋅d₀⋅d₁⋅y - 2⋅a₀⋅d₁ ⋅x\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 2 \n",
" + a₁ ⋅d₀ - 4⋅a₁⋅b₀⋅c₀⋅d₁ + 2⋅a₁⋅b₀⋅c₁⋅d₀ + 2⋅a₁⋅b₁⋅c₀⋅d₀ - 2⋅a₁⋅d₀ ⋅y + 2⋅a₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2⋅(c₀⋅d₁ - c₁⋅d₀) \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 \n",
"⋅d₀⋅d₁⋅x + b₀ ⋅c₁ - 2⋅b₀⋅b₁⋅c₀⋅c₁ + 4⋅b₀⋅c₀⋅d₁⋅y - 2⋅b₀⋅c₁⋅d₀⋅y - 2⋅b₀⋅c₁⋅d₁⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"______________________________________________________________________________\n",
" 2 2 2 2 \n",
"x + b₁ ⋅c₀ - 2⋅b₁⋅c₀⋅d₀⋅y - 2⋅b₁⋅c₀⋅d₁⋅x + 4⋅b₁⋅c₁⋅d₀⋅x + d₀ ⋅y - 2⋅d₀⋅d₁⋅x⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
"___________⎞\n",
" 2 2 ⎟\n",
"y + d₁ ⋅x ⎟\n",
"───────────⎟\n",
" ⎠"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sol_v"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
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"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.9"
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