I don't have time to delve into all that. ;-) The value in quaternions is that they are a compact, direct representation of a transformation matrix in 3D space, ergo seems ideally suited for 3D graphics abstractions. Technically, I suppose a software layer could do the optimization and map it to SIMD coprocessors, but figuring out hardware that could apply the same principles might result in even more speedup. I don't know of any algorithms with acceptable speed for quaternion multiplication, though, so regardless of whether we use existing hardware or not, it comes down to the discovery of such algorithms. Does Gaigen implement such effective algorithms?
I am not advanced enough in mathematics to know the benefits of geometric algebra, especially to graphics hardware, and thus can't speculate. I have noticed your previous postings about Geometric Algebra and do find it interesting, but struggle with figuring out how to apply it. On Tue, Jul 26, 2011 at 2:12 PM, Wesley Smith <[email protected]> wrote: > > This could change in the future to be more general purpose. For example, > > hardware-based computations using quaternions and octonions. As far as I > am > > aware, it isn't done today for purely mathematical reasons; no one knows > > how. And as far as I'm aware, such a mathematical breakthrough would be > > huge, but not something graphics vendors would pursue/fund, since it is > > "basic research" that can't be patented and so all graphics processors > would > > get the same speedup. [1] > > > Incidentally, this research has been going on for at least 10 years > already and has made significant progress in terms of compiler tools > and software systems that can be used for real-time systems. In > Guadalajara, there's a robotics group led by Eduardo > Bayro-Corrochano[1] that makes amazing machines that perform their > computations in an intrinsically spatial manner using geometric > (Clifford) algebra. One of the issues with this algebra is that the > dimensionality of the computational space grows combinatorially. The > standard 3D conformal model (5D Minkowski space) is a 32 dimensional > multi-vector. Fortunately, there's some really good software that > optimizes away the many redundancies and zero-ops called Gaigen[2], > which can handle up to 12D Clifford Algebras. Geometric algebra > subsumes quaternions and adds a lot more interesting structures and > operations. > > I don't think it requires basic research since it's just linear > algebra and easily maps to GPU hardware. Plus the research has > already been done. The software already exists for use if you want > it. I'm really not sure what interest the manufacturers would have in > it though since their more specific applications than the general case > of GA lends itself to more optimization. Also, there's an entire > world of mathematics that would have to be taught to everyone since > you aren't going to find courses in CS departments on this stuff > except in a handful of labs around the world (Netherlands, Saudia > Arabia, Mexico, Cambridge (in physics department) ...) > > > Here are some papers about GA and hardware: > > using FPGAs: > http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.1691&rep=rep1&type=pdf > conformal collision detection on GPUs: > > http://eduardoroa.3dsquash.com/portafolioeduv2/document/publications/SIACG_article.pdf > > and there are others implementing generic GA ops in graphics hardware > that I wasn't able to find as quickly. > > > [1] http://www.gdl.cinvestav.mx/edb/ > [2] http://staff.science.uva.nl/~fontijne/g25.html > > _______________________________________________ > fonc mailing list > [email protected] > http://vpri.org/mailman/listinfo/fonc >
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