Simplistic example:

I create two spheres, the larger positive, the smaller negative, in order to create a "hollow" ball.
I create two rectangles that intersect the spheres at right angles and set these to "intersect".
Group the whole thing.
The result is two circular bands, the place where the rectangles intersect the spheres.
I set THAT group to positive, cut it in half with a negative cube and I have the bands that I need for the top of the helmet started on the top left.