I have a unit sphere transformed by a given $4\times 4$ affine transform (it is a 3D transform; using homogenous coordinates with $w=1$ allows representing translation).
I now need to calculate a bounding sphere of this transformed sphere. How can I do this?
For completeness, I should mention I've thought of some bad ways to do it.
For example, one could take the sphere's axis-aligned bounding box, transform that, and then compute the minimal bounding sphere of those 8 points. This is unsatisfactory because the bounding sphere will be very coarse. E.g. if the transform were the identity, we can see that the new bounding sphere will have $3 \sqrt{3} ~\times$ the volume.
So as additional requirements: the preferred method will produce a fairly tight bounding sphere and also be reasonably simple ($O(1)$ time, small constant).