CODEWITHSUNDEEP
pythonsympy
Aurelie Navir
Aurelie Navir

Reputation: 1008

Solving this equation in sympy

I'm playing with rotational matrices and similar concepts using sympy. I'm trying to extract alpha, beta, gamma from the following matrix:

⎡       α - γ⋅sin(θ)        ⎤
⎢                           ⎥
⎢β⋅cos(φ) + γ⋅sin(φ)⋅cos(θ) ⎥
⎢                           ⎥
⎣-β⋅sin(φ) + γ⋅cos(φ)⋅cos(θ)⎦

so that I get this matrix

⎡1     0        -sin(θ)   ⎤
⎢                         ⎥
⎢0  cos(φ)   sin(φ)⋅cos(θ)⎥
⎢                         ⎥
⎣0  -sin(φ)  cos(φ)⋅cos(θ)⎦

Basically solve the equation F(x) = Ax, but for A matrix instead of x. Where F(x) is the first matrix, A the second matrix and x is the alpha, beta, gamma vector. I naively tested with sympy.Solve but it outputs Matrices A and x are not aligned. From what I've seen usually is because the dimensions of the matrices are not correct but in my case I think it's because using solve is not the correct way.

How could I solve this with simpy?

Upvotes: 0

Views: 179

Answers (2)

Oscar Benjamin
Oscar Benjamin

Reputation: 14530

You can use SymPy's linear_eq_to_matrix function:

In [6]: a, b, g = symbols("α β γ", positive=True)
   ...: t, p    = symbols("θ φ",   positive=True, nonzero=True)

In [7]: bres = Matrix([[a - g*sin(t)],
   ...:                [b*cos(p) + g*sin(p)*cos(t)],
   ...:                [-b*sin(p) + g*cos(p)*cos(t)]])

In [8]: bres
Out[8]: 
⎡       α - γ⋅sin(θ)        ⎤
⎢                           ⎥
⎢β⋅cos(φ) + γ⋅sin(φ)⋅cos(θ) ⎥
⎢                           ⎥
⎣-β⋅sin(φ) + γ⋅cos(θ)⋅cos(φ)⎦

In [9]: linear_eq_to_matrix(bres, [a,b,g])
Out[9]: 
⎛⎡1     0        -sin(θ)   ⎤  ⎡0⎤⎞
⎜⎢                         ⎥  ⎢ ⎥⎟
⎜⎢0  cos(φ)   sin(φ)⋅cos(θ)⎥, ⎢0⎥⎟
⎜⎢                         ⎥  ⎢ ⎥⎟
⎝⎣0  -sin(φ)  cos(θ)⋅cos(φ)⎦  ⎣0⎦⎠

https://docs.sympy.org/latest/modules/solvers/solveset.html#linear-eq-to-matrix

Upvotes: 2

evilmandarine
evilmandarine

Reputation: 4553

This is the closest I could come up with. As you know F(x) (called bres below) and x, you're looking for A matrix which are 9 variables (a11, a12, etc.). Multiplying A by x gives you 3 equations with independent variables. We know Ax - b = 0, but to find each coefficient you need to add constrains. For example to find a11, you need to solve the first equation for a11 but as you're looking for the α coefficient, you need to add the constraint that β and γ are 0 (subs function).
Please also see my comment to your answer, as this is probably not the best way to solve it.

from sympy import *

a, b, g = symbols("α β γ", positive=True)
t, p    = symbols("θ φ",   positive=True, nonzero=True)

x = Matrix([[a],
            [b],
            [g]])

bres = Matrix([[a - g*sin(t)],
               [b*cos(p) + g*sin(p)*cos(t)],
               [-b*sin(p) + g*cos(p)*cos(t)]])

a11, a12, a13, a21, a22, a23, a31, a32, a33 = symbols("a11, a12, a13, a21, a22, a23, a31, a32, a33", real=True)
A = Matrix([[a11, a12, a13],
            [a21, a22, a23],
            [a31, a32, a33]])

v = A*x-bres

Ares = Matrix([[solve(v[0].subs(b, 0).subs(g, 0), a11), solve(v[0].subs(a, 0).subs(g, 0), a12), solve(v[0].subs(a, 0).subs(b, 0), a13)]
              ,[solve(v[1].subs(b, 0).subs(g, 0), a21), solve(v[1].subs(a, 0).subs(g, 0), a22), solve(v[1].subs(a, 0).subs(b, 0), a23)]
              ,[solve(v[2].subs(b, 0).subs(g, 0), a31), solve(v[2].subs(a, 0).subs(g, 0), a32), solve(v[2].subs(a, 0).subs(b, 0), a33)]])

print(Ares)

Result:

Matrix([[[1], [0],       [-sin(θ)]],
        [[0], [cos(φ)],  [sin(φ)*cos(θ)]],
        [[0], [-sin(φ)], [cos(θ)*cos(φ)]]])

Upvotes: 1

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