[FIXED] Three spheres intersection (trilateration) with SymPy

Issue

Is there a way to get a solution to three spheres intersection (trilateration) with SymPy? sympy.geometry doesn't have a sphere object, so a direct approach is not feasible. Can SymPy solve a system of non-linear equations as shown at Trilateration and the Intersection of Three Spheres?

Solution

Yes. There are different ways but e.g.:

In [21]: x, y, z = symbols('x, y, z', real=True)

In [22]: eq1 = (x-1)**2 + (y-2)**2 + (z-3)**2 - 1

In [23]: eq2 = (x-1)**2 + (y-S(5)/2)**2 + (z-3)**2 - 1

In [24]: eq3 = (x-S.Half)**2 + (y-S(5)/2)**2 + (z-3)**2 - 1

In [25]: solve([eq1, eq2, eq3], [x, y, z])
Out[25]: 
⎡⎛              √14⎞  ⎛          √14    ⎞⎤
⎢⎜3/4, 9/4, 3 - ───⎟, ⎜3/4, 9/4, ─── + 3⎟⎥
⎣⎝               4 ⎠  ⎝           4     ⎠⎦

Answered By - Oscar Benjamin