I would like to plot the possible solutions (x, y, z) of the following equation system: dn^2=(x-xn​)^2+(y-yn)​^2+(z-zn)^​2 where n=1 to 3 and dn, xn, yn, zn are known

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Liviu on 22 Mar 2023
Commented: Torsten on 25 Mar 2023
d1^2=(x-x1)^2+(y-y1)^2+(z-z1)^2
d2^2=(x-x2)^2+(y-y2)^2+(z-z2)^2
d3^2=(x-x3)^2+(y-y3)^2+(z-z3)^2
d1,d2,d3 are known (say 100, 110, 120)
x1, x2, x3, y1, y2, y3, z1, z2, z3 are known, say 50, 50, -50, 75,70,65, 5,-5,10

John D'Errico on 22 Mar 2023
You have 3 spheres, with known centers and radii.
Expand those equations. Then subtract them from each other. So maybe Eq1-Eq2, Eq2-Eq3, and Eq3-Eq1.
What happens to the x^2, y^2, z^2 terms? They go away. You are left with three simultaneous LINEAR equations in three unknowns. Solve that for x,y,z. That is the solution for the intersection of the three spheres. Nothing complicated.
Torsten on 25 Mar 2023
@Liviu comment moved here:
Thanks! Yes, there might be 3 sphres but there are multiple possible values that saisfy the system. The problem is not how to solve it but how can I visualise thesevalues, within some sort of 3D graph. I am new in Mathlab and I don't know the functin that will create this 3D plot...BTW, the solution can be obtained using "Solver" in Excel and proper condiitons...

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