Displaying only part of an object

I want to have some objects that aren't displayed until an other object intersect them and then they are visible only for the intersected part, something like this (but in 3D):

where the blue part is the part to be shown and the circle is the moving area that make the objects appear.

Searching around, I found this, let's say that what i want is the opposite.

How could I achieve such effect?

Thanks.