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I am currently working my way through some geometric algebra (as well as algebraic geometry and topology) texts and building a [programming] library incorporati
by Dn_Ab 15y ago
I am currently working my way through some geometric algebra (as well as algebraic geometry and topology) texts and building a [programming] library incorporating what I read to reinforce my mathematical intuition. I completely agree about how awesome Geometric Algebra is. In fact its emphasis of tying algebra with geometry makes it much more useful than for just teaching geometry. It makes a wonderful foundation for mathematical intuition. For self study I particularly like such subjects because they take a whole bunch of stuff and put them under one roof.
Grassmann's story is semi-tragic. He was a school teacher who created a powerful algebra of space where you could compute with not just points and vectors but with lines, planes, surfaces and other vector subspaces without being hampered by a particular coordinate system. Geometric Algebra can subsume complex numbers, quaternions and plane geometry - projections, rotations etc and easily extend them to higher surfaces. Matrices play less a role here. Hestens also puts forward that Geometric Calculus can contain matrix calculus, linear algebra, lie algebra and groups and Differential forms, see [1]. But because Grassmann was so ahead of his time and did himself no favours by writing a dense, super long text with lots of philosophy interwoven not many bothered to read or could understand it. Later, Clifford came and made extensions, cleaned it up and the subject gets named after him as Clifford Algebras, though not by his choice. But Clifford died early so the more geometrically unwieldy Vector Analysis won out. A setback for physics?...Angular momentum makes so much more sense as a bivector. From [2]:
Fearnley-Sander writes in [27]:-
All mathematicians stand, as Newton said he did, on the shoulders of giants, but few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.
To read more about Grassmann I recommend: A History of Vector Anaylsis by Michael Crowe, well written, I couldn't put it down.
[1] http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf
[2] http://www.gap-system.org/~history/Biographies/Grassmann.html http://www.gap-system.org/~history/Biographies/Grassmann.htm...
- copper 15y agoPerhaps the most interesting part, to me, was the goal of making geometric algebra operations computationally efficient. While I'm still reading (and enjoying) the original paper, I must confess an inability to see how it's more efficient than a standard vector representation. Would you have any pointers on that?
- Dn_Ab 15y agoAt the moment, the core of the gains are in the conceptually clean framework at the cost of speed. Kind of like going from C to Python. It will help your linear algebra by strengthening your geometric intuitions, stuff like how to figure out if a plane intersects a sphere or if a line intersects a plane or SLERP are much are easier for the novice. You solve it algebraically but the objects you are reasoning with are geometric entities so you can't help but build your intuition on both. Computationally though, they take more operations and elements (2^dim). And there is an order of magnitude difference in speed - 100x if implemented naively. But there are optimizations that can get it to 2x - 7x [1]. But apparently for full generality, 3D geometry is best done in a 5D conformal geometry which requires 32 element representations but I haven't gotten that far yet. You can make it more efficient by taking advantage of the sparseness of most structures and being able to scale up to more information as required by filling more indices. Something that cannot be handled as cleanly in standard vector based methods. Strangely this method of conformal geometry is covered by a Patent so commercial usage in the US requires licensing arrangements [2]. [1] http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compressed.pdf http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compr... [2]http://www.google.co.uk/patents?id=0CkWAAAAEBAJ&pg=PA7&dq=System+for+encoding+and+manipulating+models+of+object&hl=en&ei=RYbTTo6jK8LR8QOssMn9Dw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDcQ6AEwAA http://www.google.co.uk/patents?id=0CkWAAAAEBAJ&pg=PA7&#...
- copper 15y agoThanks, that's interesting, and seems more in line with what I figured: a lot easier to reason about, and a much cleverer coder than me to do it properly :) (As an aside, one of the inventors listed on that patent happens to be the person who wrote this article.)
- jacobolus 15y agoThat Fearnley-Sander paper is here, by the way: http://mathdl.maa.org/images/upload_library/22/Ford/DesmondFearnleySander.pdf http://mathdl.maa.org/images/upload_library/22/Ford/DesmondF...