Scientific visualization now lies somewhere between a growth industry and a buzz word. It's become an important tool for the scientific research community, is rapidly becoming one in undergraduate education, and has captured the imagination of the (semi)popular press. There are a lot of mathematicians involved in GEOMETRIC visualization, yet I don't know MATHEMATICIANS who know much about the mathematical content of SCIENTIFIC visualization. And I think this could be important, as I'll explain below.
My guess is that scientific visualization uses --three-dimensional analytic geometry (but what?), --occasional pieces of other areas of classical mathematics, such as differential geometry, --new geometry, in particular computational geometry, --multivariate statistics, plus things I just don't know about.
WHY THIS COULD BE IMPORTANT
--Three-dimensional geometry has just about disappeared from the college and from the school curriculum. We've all noticed the "splat" students make when they bump into having to visualize simple three-dimensional objects in several variable calculus. I'd argue that some parts of three-dimensional geometry are very important for the college curriculum, but I'm not sure exactly what parts. It might be useful to look at what is needed in scientific visualization. Certainly I'd expect the new geometric visualization software being developed at the Geometry Center and other places might be very helpful in deciding how to teach three-dimensional geometry. This time around, we might even be able to make it an interesting course!
--Moreover, we just might find some interesting mathematics we should be teaching future scientists, and we might find some cogent arguments for the mathematics we're now teaching.
--There are at least some instances of fruitful collaboration between mathematicians and scientists(!) in scientific visualization, and I think this deserves to be more widely known.
--At least some areas of scientific visualization appear to be fertile breeding grounds for new mathematics. We should be preparing students for this. (Incidentally, has computational geometry, our most recent child, been completely abandoned to its foster-parent computer science, or is it still nurtured in some math departments?)
--In general, some of the mathematics used in scientific visualization may be important for mathematics and for mathematics education, and this mathematics may not yet be recognized as important.
The Media Magic catalog (mentioned several times before on the Forum) has, among other things, an imposing list of books on scientific visualization, none of which are in our local library. The prices are imposing, too, but I've just been given some unattached research funds which were equal to the task. I plan to look through a batch of books and let you know what I find of interest relating to the above. "Review" would be too strong a word, since I won't have time to examine them in real detail, but perhaps I can inspire others to take a more careful look and report their findings.
Brodlie, K.W. et al (Eds.), "Scientific Visualization Techniques and Applications, Springer-Verlag, '91
Kaufman and Nielson, (Eds.), "Proceedings of Visualization '92", IEEE Press, '92
Keller and Keller, "Visual Cues", '92
Patrikalakis,N.M. (Ed.), "Scientific Visualization of Physical Phenomena", Springer-Verlag, '91
Thalman, Daniel (Ed.), "Scientific Visualization and Graphics Simulation", Wiley, '90
There are other books available, but from two-paragraph descriptions these appeared to have the best chance of having at least a little bit of mathematical content, and not be entirely gee-whiz pictures.
WHAT I'D LIKE FROM YOU
Help! Do you know or know people who know anything about the mathematical content of scientific visualization? Would you like to look over and report on a book? Do you know other important sources which should be considered? Etc?
Also, please do comment about this project (how's my jousting technique? have I picked a good windmill?), and on the book reports I'll soon be giving. What could I do to make the project more worthwhile?