41–50 of 184 posts
Basicallt, quaternions are hard to understand but easy to use.
Texts on GA seem to start out saying "we're going to replace vector analysis with something intuitive and natural and eloquent and..." and then immediately introduce the "geometric product", which is neither intuitive, eloquent, or, as far as I can tell, natural. Its properties are found via formula-wrangling, and the results you get from it seem to just magically work, rather than being intuitive. Bivectors in gener…
Hmm, the word "interpolate" does not appear in the article. One of the main applications of quaternions in games is interpolating rotations. Does this representation interpolate well? Also, I've seen several articles trying to convince me that alternative transformation representations like these are better, but I haven't seen much code. Has anyone written a library using these concepts that could replace a tradition…
Yes, it does, because it is an identical representation to quaternions. Except it actually explains all the weirdness in quaternions without resorting to 4-space. This is because quaternions are a subalgebra of geometric algebra in 3-space.
This article will take more than than I have to digest right now, but it's definitely saved. I have never really understood quaternions. All I knew is that I could use that instead of euler angles, avoid gimbal lock, and would blindly use them.
I think you just described 90% of people who use them. Normally I like having a better grasp of the underlying theory I'm applying, but somehow quaternions seem more likely to make me think, yeah I'll eventually get around to that... instead of diving right in as this article I believe better motivates.
Exactly. And then you forget about them because...well, it's not like this stuff requires maintenance. Once you abstract all rotations and interpolations away, you just use them and forget you don't really grasp why they work.
geometric algebra proponents are the rust strike force of mathematics
When just implementing a first person type control/viewport I found it way easier to just use euler angles and the gimbal lock didn't really feel unnatural to me. So I think there is something to the idea that maybe you don't even need the complexity of quaternions at all dependent on the usecase.
I do love Geometric Albegra and hope it gets further adoption. In my field (physics), the project of moving to geometric algebra currently looks hopeless due to the inertia of other formalisms and lack of interest from established physicists. The real shame is that the best opportunity to introduce physicists to geometric algebra was a hundred years ago when we were first discovering spinors. First the Pauli algebra…
For most of the usecases associated with the examples you mention - Pauli and Dirac algebras - I don't think it matters very much. For instance, I haven't seen a decent QM course that doesn't devote at least a lecture or two to discussing the origin of Pauli matrices and the associated abstract algebra and group theory - as they should - but I don't think the course would be any more concise or useful (in terms of the physical content) if it then proceeded to treat QM in the language of geometric algebra. There's nothing wrong with the matrix representations (especially in a more computational context), though obviously one should know where the representation comes from and how to derive it.
I read through a bunch of this, and one part seems poorly explained. There’s a clear definition of the product of two vectors, but the result isn’t a vector — it’s a scalar plus a bivector. Then, by magic, three vectors get multiplied! But what does that even mean? Somehow a vector gets multiplied by a bivector, and the result is somehow still a vector. What’s going on?
You can work everything out by breaking it down to the unit basis vectors, for which the products are explicit. One way of writing it, which feels more comfortable to me possibly at the expense of being the "wrong" sort of intuition, is
u(v^w) = (u.v)w - v(u.w) + u^v^w
(using . for inner product and ^ for outer product). I guess you can also think of the first two terms as the rule for taking the inner product of a vector and a wedge product, though I haven't thought through this completely.
Because of the repeated vector in the reflection formula, that last trivector vanishes, so the result is just a vector, as it will be whenever the vector is coplanar with the bivector.
Texts on GA seem to start out saying "we're going to replace vector analysis with something intuitive and natural and eloquent and..." and then immediately introduce the "geometric product", which is neither intuitive, eloquent, or, as far as I can tell, natural. Its properties are found via formula-wrangling, and the results you get from it seem to just magically work, rather than being intuitive. Bivectors in gener…
Let {e1,e2} be an orthonormal basis for R^2.
Let i = e1e2.
Then i^2
= e1e2e1e2
= -e1e1e2e2 (anticommutivity)
= -(1)(1)
= -1.