Q:

Can the intersection of two balls be described?

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 ?

A:

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 $r_i$.

