Thalmann, Daniel (Ed.), "Scientific Visualization and Graphics Simulation", Wiley, '90, $50, 264 pages.
"The purpose of this book is to present the techniques of computer graphics, image synthesis, and computer animation necessary for visualizing phenomena and mechanisms in sciences and medicine". It's based on lectures given at the Computer Graphics Laboratory of the Swiss Federal Institute of Technology in Lausanne. It is a reasonably well written, indifferently proofread collection of good survey articles. I'll only speak about the parts which strike me as having interesting mathematics, and I shan't grouse about those parts which don't, since the book wasn't really written with us in mind.
Chapter 2 on algorithmic (computational) geometry briefly treats several fundamental problems which frequently arise: intersection problems, location problems, convex hulls, and proximity problems.
Chapter 3 on surface visualization. What is a surface and how is it stored? Level surfaces: unstructured or structured discrete or parameterized. How to draw a surface and how to recognize its shape. More computer graphics heuristics than mathematics. A mention of differential geometric tools. For differential geometry, we're referred to Nutbourne, A.W. and Martin, R.R. "Differential Geometry Applied to Curve and Surface Design", Ellis Horwood, Chichester, '88, and to Faux, I.D. and Pratt, M.J. "Computational Geometry for Design and Manufacture", Ellis Horwood, Chichester, '79. Does anyone know these books? We don't have them locally, and I'd like to know what's involved in applied differential geometry.
Chapter 4: Solid Modeling. Mentions the use of classical algebraic topology to decide what should be a "valid object" in solid modeling, and how it should be represented. Apparently, a seminal reference is Requicha,A.A.G. "Mathematical Models of Rigid Solid Objects", Technical Memo TM-28, University of Rochester, '77. Anybody ever looked at it? Euler's Formula can be used to keep track of "topological validity", but apparently is expensive for computers.
The chapter also considers "Constructive Solid Geometry", but I didn't quite get what it is, or find references beyond an article by Laidlaw and Trumbore in SIGGRAPH, '86. Hints of interesting mathematics in the background. How 'bout "Computational Topology", anyone? (I'll provide the reference if you'll try to find and report on it).
Chapter 5 is on the Finite Element Method. This is an ancient differential equations technique, and this naive reader emerged from a few minutes immersion in the chapter none the wiser as to how it's applied to scientific visualization, or as to what (if anything) it has to do with my quest. Enlightenment, anyone?
Chapter 6: Visualizing Sampled Volume Data, is mostly devoted to standard 3-dimensional techniques, although the author briefly mentions higher dimensions, and acknowledges the need for imaging methods that can show multiple parameters. We learn that many volume renderers use parallel projection, because it's cheaper, and for scientific data it may be preferable to not alter size with distance. However, some renderers do offer perspective projection, which is better for depth perception. Sensory input other than visual can help dealing with multiple parameters. For example, the frequency rate of repetition of a sound could be used to indicate density as the user moves through the environment [Grinstein, G., Pickett, R.M.,. Williams, M.G. "Exvis: an exploratory visualization environment", Proceedings Graphics Interface '89, pp.254-261, June '89]
Chapter 7, Special Models for Natural Objects, starts with a brief discussion of fractals, with some emphasis on Barnsley's Iterated Function Systems. There is a description of structural modeling of botanical structures.
Chapter 8 seems a perfectly fine introduction to Computer Animation, but there's little relating to our quest.
Chapter 10 on Robotic Methods for Task-level and Behavioral Animation is concerned with robotics techniques which are used or could be used for the task-level animation of linked figures. In direct kinematics, position and velocity are transformed from "joint-space" into cartesian coordinates. Other methods are based on Lagrange's equations for motion for nonconservative systems, the Gibbs-Appel equations, or on the Newton-Euler equations. I was expecting more computational geometry, but it might have passed me by.
The theme of Chapter 11, Visualization of Flow Simulations, seems to be the necessity of interaction between software and a human with knowledge of what the solution should be like. Many aspects of computational fluid dynamics require massive computing power.
Chapter 12 on the visualization and manipulation of medical images, doesn't have much of overt mathematical interest. Apparently the most widely used method of measuring left ventricular volume is to model the ventricle as a three-dimensional ellipsoid. Some things never go out of fashion.
Chapter 13, the Visualization of Botanical Structures and Processes, is concerned with Lindenmayer systems, a "topological" description of the modeled structure, which is drawn by a LOGO-like turtle. Phylotaxis is given by logic-like production rules, with tiny bits of cylindrical coordinates, etc. The chapter manages to avoid the two or three neat things I've picked up about the subject--guess they weren't of practical importance.
Chapter 14 on molecular computer graphics in chemistry has little overt mathematics, nor does Chapter 15: Graphics Visualization and Artificial Vision, or Chapter 16 on Collaboration Between Computer Graphics and Computer Vision.
Chapter 17 on Graphics Simulation in Robotics has comments and references of mathematical interest, although frequently in the form of laments that easy problems lead to surprisingly difficult mathematics. For collision detection, it is sometimes convenient to treat intersection problems as occurring in 4-space.
It's interesting that the background assumed for users of scientific visualization systems seems to vary from zero in medical imaging for routine clinical evaluations to a good understanding of the solutions to the partial differential equations governing fluid dymanmics in flow simulation.