Doubt in Sphere Sphere intersection

Question

I am trying to understand the sphere-sphere intersection in http://www.ambrsoft.com/TrigoCalc/Sphere/TwoSpheres/Intersection.htm

I have a doubt. Can you tell me is AB(diameter of common circle) is perpendicular to line joining centers? Even wolfram takes that common plane is perpendicular to line joining centers....(http://mathworld.wolfram.com/Sphere-SphereIntersection.html)

Kindly enlighten...

Answer 1

Can you tell me is AB(diameter of common circle) is perpendicular to line joining centers?

The common circle is the solution of the system: $$ x^2 + y^2 + z^2 = r_1^2 \\ (x-d)^2 + y^2 + z^2 = r_2^2 $$ Assuming $d > 0$, for the solution points $(x,y,z)$ we get the constraint $$ -2xd+d^2=r_2^2-r_1^2 \iff \\ x = \frac{r_1^2 - r_2^2 + d^2}{2d} = x_m = \text{const} $$ by subtracting the first from the second equation and solving for $x$, so we have a circle in the $y$-$z$-plane, perpendicular to the line which connects the two center points.

We would get the circle equation by substituting the above expression into the first circle equation and then sorting the constant terms to the right, getting at some equation $y^2 + z^2 = R^2$ together with $x=x_m$.

Answer 2

Note that the line joining the centers is the perpendicular bisector of $AB$ because of the symmetry of each circle.

Thus $AB$ is perpendicular to the line joining the centers as well.