Center coordionates of the ring

Question

If you have two points on the circumference of the ring having coordinates p (x1,y1,z1) and q (x2,y2,z2) radius and angle not known. How can we get the coordinates of the center of the ring

Answer 1

We need 4 equations, as Pc=(Xc, Yc, Zc) is the center, and radius are all unknown:

You can find the distance between P and C

and

the distance between Q and C

where C=(Xc,Yc,Zc) where C is the center.

since they are on the circle:

(x1 - xc)^2 + (y1-yc)^2 + (z1-zC)^2 = R^2

and

(x2-xC)^2 + (y2-yC)^2 + (z2-Zc)^2 = R^2