Circles can also intersect perpendicularly in a more complicated way than discussed in Spheres IV. Like so:
This might look complicated, but is in fact just a transformed version of the easier to grasp dart disk:
To see how these two images are related, pretend the radial lines in the second image are in fact huge circles that all intersect in the center point. Then they will also intersect in another point, which is, in the case of lines, the ominous point at infinity, but, in the case of circles, becomes just another point in the plane. This other point and the origin are the common points of one family of circles, as you can see in the first image, and the second family of circles intersects the first perpendicularly. The first image can be transformed into the second by what is called an inversion.
If we want to repeat this in three dimensions, it is maybe best to start with the second image, replacing the radial lines by vertical green planes, and the circles by concentric blue spheres. Then, something curious happens. Lines and circles are in some sense the same thing, and so are planes and spheres. But if we look for a third family of surfaces that intersect the planes and spheres orthogonally, we need to step outside the plane/sphere paradigm. It turns out that we need vertical red cones to cut both the blue spheres and the green planes perpendicularly:
Now, coming back to the 3D version of the first image, we just need to invert the above cones, planes, spheres as to become this:
The red surface is called a cyclide. It has two cusps that correspond to the tip of the cone and the (still ominous) point at infinity.
Now imagine that you are inside that cyclide, looking around…