181–184 of 184 posts
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This is closely related to the exterior algebra, which is where bivectors live. But I don't know much about Clifford algebras either. I didn't know that they are more directly connected to Quaternions. Thanks!
The 'geometric product' mentioned in the article is just Clifford's product. The 'geometric algebra' is the same thing as the Clifford algebra induced by the scalar product.
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But if the argument is "let's replace quaternions in 3D engines!" then the question is really more "Okay, so do they outperform quaternions, then??" because if you're writing the 3D engine, the underlying code only matters insofar as that it has to do what it's supposed to do. You don't need to understand why it works, just that known operations will lead to known intended results. And if you're _using_ a 3D engine,…
> But if the argument is "let's replace quaternions in 3D engines!" then the question is really more "Okay, so do they outperform quaternions, then??" because if you're writing the 3D engine, the underlying code only matters insofar as that it has to do what it's supposed to do. You don't need to understand why it works, just that known operations will lead to known intended results. At some point, people will have t…
Is "what they do" unintuitive? Sure, but I couldn't care less about whether someone can conceptualise _why_ they do what they do, I care about _that_ they do what they do, and that I can trust them to do that. If someone can't understand what geometric operation an non-geometric transformation might map to "in between the start state and the end state" then that someone is focussing on the wrong thing twice:
1. quaternion maths is not geometry, it's algebra. It has some nice geometric analogies when performing specific operations, but trying to understand them _as_ geometry and expecting them to make sense across the board is ridiculous, and 2. if you feel you need to understand the "intermediary" results of a 4 dimensional algebra before you're willing to write or maintain code that uses it, maybe you're putting your foot down in the wrong spot: this is a 3D engine. It's not "easy code that we over-complicated by using quaternions", it's stupidly complex code that we simplified by working with quaternions.
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Can I ask you (and keldaris) what precisely you mean by geometric algebra, and how you would propose to teach it? I'm a math professor. To me "geometric algebra" refers to any use of geometry in mathematics to illuminate algebraic principles. There are very very many of these -- this article illustrates one good example. I try to incorporate intuition from geometry into my teaching whenever possible. It seems that th…
Geometric algebra has a definition that is more specific than any use of geometry to teach Elementary Algebra on real numbers. https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra For teaching geometric algebra, there are curricula and textbooks available... http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ https://arxiv.org/abs/1205.5935v1 http://www.faculty.luther…
This looks a bit too high-level to inform my undergraduate teaching, but certainly I could learn something from this.