Suppose, two spheres intersect. Subtracting the equations of the speheres, a linear equation appears which indicates the plane conataining all points belonging to the intersection of the spheres.
But the intersection is only a small part of this plane.
- Can this intersection be described ? Do the points form a well-known geometric shape ?
You can always choose a coordinate system such that the first sphere is the larger one (if radii differ) and is centered around the origin: $$ x^2 + y^2 + z^2 = R^2 \quad (*) $$ and that the second sphere is centered around $(0,0, h)$: $$ x^2 + y^2 + (z - h)^2 = r^2 \quad (**) $$ for an intersection we should have $h \le R + r$.
For the intersection both equations $(*)$ and $(**)$ must hold. Substracting $(*)$ from $(**)$ we get:
$$ -2zh+h^2 = r^2-R^2 \\ z = \frac{h^2 + R^2 - r^2}{2h} =: z_i $$
The equation $(*)$ then reduces to $$ x^2 + y^2 = R^2 - z_i^2 = r_i^2 \quad (\#) $$ with $$ r_i = \sqrt{R^2 - z_i^2} $$ The equation $(\#)$ of the intersection is the equation of a circle with radius $r_i$.