How to find one of the points of a triangle $3$d?

Question

How to find the coordinates of $C$ as a function of $A, B, d_1, d_2$ and $d_3$?

$$f(A, B, d_1, d_2, d_3) = C$$

I need to find the coordinates of point C as a function of the distances d1, d2, d3.

A(0,0,0)

Answer 1

C is not uniquely defined by the conditions you are mentioning simply because if we assume $C(x, y, z)$ then we have the following constraints

$$(x-A_x)^2+(y-A_y)^2+(z-A_z)^2={d_1}^2$$ and

$$(x-B_x)^2+(y-B_y)^2+(z-B_z)^2={d_2}^2$$

Both of them are spheres and hence they intersect in a circle. Thus locus of C is a circle of intersection of the above two spheres assuming that the initial constants are well chosen and they define a real triangle.