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Comment by copper | Hacker News Reader

Perhaps 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?

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At 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_compr...

[2]http://www.google.co.uk/patents?id=0CkWAAAAEBAJ&pg=PA7&#...

Thanks, 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.)

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