In extending the tubes down toward the (x,y)-plane,
we will proceed one step at a time.
On the first step, we will go from height
1 to height 1/2,
on the second step we will go from height
1/2 to height 1/4,
etc.
On the kth step we go from height 
down to height 
.
As we do this, we will only need to consider
hemispheres of radius > h / 4,
since these are the only ones that protrude into the region
we are cutting up.
So let us define the
relevant hemispheres
to be those hemispheres
(other then 
and 
)
whose radius is > h/4.
(See Figure 10.)
Figure 10: Relevant hemispheres.
Note that we do not consider a hemisphere relevant if its radius is exactly h/4, so that the apex of its hat is at height exactly h/2. Of course which hemispheres are relevant depends on which step we're working on.
It is precisely because we only have to consider big hemispheres that this one-step-at-a-time approach will allow us to avoid the confusion that we anticipated from sending tubes starting above one hemisphere out over other hemispheres. There may well be other hemispheres beneath where we send our tubes, but they're irrelevant because they're small and don't get in the way. When we're down so that our distance to the plane is comparable to their radius, then we'll worry about those little hemispheres.