From the write-up for the book;
"...with the key concepts of quadrance and spread replacing distance and angle."
It sounds like quadrance is distance. If it is the same as a^2 + b^2 (without the square root), then it sounds familiar. People doing computer graphics often need to compare distances, which can be done more efficiently by not taking the square root. This is nothing new.
I read the book write-up(hype), but was left with the question:
What's the problem?
Also, my impression is that this is really not something new - maybe its formalism, but it sounds much like what people doing computer graphics or geometric modeling do to avoid calculating trigonometric functions. Everything in computer graphics has to be optimized and I have 5 books on my shelf (Graphics Gems I-V, 3000 or so pages) that go into these short-cuts in detail. Just like there are explicit, implicit, and parametric equations, different approaches have different trade-offs.
Trig functions really aren't difficult, so what's the problem?
-- SteveH - 20 Sep 2005