geometric problems with interval arithmetic

[David G. Hough at validgh]

Vaughan Pratt observes that rectangular intervals are usually suboptimal as
dimensionality increases, and this is certainly true, as I've commented before
in the context of orbit problems.   You want to bound general convex sets,
and realistic bounds may be too costly.     Kahan's approach on orbit 
problems was to use ellipsoids rather than parallelopipeds, but Bill Walster
has told me that there is recent progress among interval researchers in 
obtaining realistic growth in error bounds for orbit problems using only
conventional rectangular intervals, and more generally the good news about
interval methods is that they are being brought to bear to solve realistic
problems, economically and with guaranteed results, more often than in earlier
years.

If all interval operations were complex multiplication and division, then
the optimum interval arithmetic would be polar coordinates - I think the
product or quotient of polar-coordinate intervals is another polar-coordinate
interval, the obvious one.   Things get complicated if you want to add
and multiply though.    One of the early approaches to interval methods 
still used from time to time is to represent them as a ball by center
and radius - using less storage than rectangular coordinates - but this 
approach has its limitations too.