Thursday, August 03, 2006


I only attend one conference a year, SIGGRAPH, the annual graphics convention.

I think that over the years the complexity of the mathematics used in graphics has been creeping upwards and I think the field is moving into a more mature phase. However, there are usually very few results that are of purely mathematical interest. One area that does look like it might be of general mathematical interest was the course on discrete differential geometry (DDG). Unfortunately I didn't actually attend the course but I can say a tiny bit on why this might be interesting. (This is all second hand from people who did attend the course.)

In many graphics applications (including physical simulation) we have to use calculus on manifolds. For example when solving the Navier-Stokes equations for fluid dynamics or for smoothing the surfaces of 3D models. Unfortunately, in graphics we tend to work with triangulations of surfaces and use discrete approximations to derivatives. As a result, we can only use our usual theorems of calculus to make approximate statements about these triangulations (usually based on some form of finite differencing).

The DDG approach appears to be an alternative well principled approach to these derivatives. Although they still only form approximations to the continuum limit, discrete analogues of the usual theorems of calculus hold exactly. For example it is possible to define discrete versions of differential forms and the exterior derivative leading to a discrete version of de Rham cohomology.

One place where DDG appears to perform well is in fluid dynamics. In the graphics world we don't care about how accurate these simulations are. What we need is for simulations to be stable and be plausible. Often standard approximation methods lead to effects like mass loss and vorticity disspipation. These quantities are often salient to the eye. The effect may be small but over a period of time the problems can accumulate and it can soon become apparent that simulation has errors. What's cool about DDG is that discrete analogues of the usual conservation equations are provable. As a result, even though the simulation is an approximation to the continuum, the discrete analogues of the conserved quantities remain good approximations to the continuum values. Even though the simulation may end up being inaccurate, it remains looking good.

The other good thing about SIGGRAPH is the publishers selling heavily discounted books. (Bonus discounts if you've published a paper with them.) Among other books I picked up paper copies of Generatingfunctionology, Synthetic Differential Geometry (both available in electronic form, but I like paper), Knuth's History of Combinatorial Generation (I was aware of Abulafia's interest in combinatorics (though with Madonna's recent interest in Jewish Mysticism I ought to distance myself from that) but not the many other developments from other cultures) and this book which almost brought tears of nostalgia to my eyes.

By the way, I have two free days in the Boston area. Anyone have any suggestions for what to do here? I'm already planning to (re)visit the MIT museum and some of the art galleries. Any other suggestions?


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