This article reads like 'I don't understand quaternions therefore they're bad.'
Like what - they are the objectively best representation of rotations...
Let's remove Quaternions from every 3D Engine
51–60 of 184 posts
Re: Let's remove Quaternions from every 3D Engine
#52It's completely unnecessary to understand what the individual components of quaternions represent. All that is necessary is to build an understanding of what the operations do for you. Do you need to understand Russell's proof that 1+1=2 to compute the tip on your bar bill, or do you just need to know what to expect when you apply the addition operation to two numbers? Treat quaternions as indivisible whole values as you would floating point numbers.
Re: Let's remove Quaternions from every 3D Engine
#53Texts 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…
It's very cool that the geometric product of any orthonormal basis for R^2 gives i as a "pseudoscalar". Let {e1,e2} be an orthonormal basis for R^2. Let i = e1e2. Then i^2 = e1e2e1e2 = -e1e1e2e2 (anticommutivity) = -(1)(1) = -1. Personally that alone is enough to justify learning more about geometric algebra.
That is true of the wedge product also, though. That's my point. The geometric product gives very little power over the wedge product, and when it does, it's massively non-intuitive how it works.
Re: Let's remove Quaternions from every 3D Engine
#54I thought Quaternions solved discontinuities in rotations.
Re: Let's remove Quaternions from every 3D Engine
#55I 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…
Do you think nature requires such an exotic representation?
I would argue that there is nothing exotic about geometric algebra and that its a much more natural representation of nature than our current hodge-podge of traditional vectors, tensors, complex numbers, differential forms and spinors.
Re: Let's remove Quaternions from every 3D Engine
#56I 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…
In my personal experience physicists are usually rather pragmatic about these things. Not many will support a "project of moving to geometric algebra" for its own sake, or the sake of perceived mathematical elegance - I know I wouldn't, even though I spent some time studying it just for fun. On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few…
> On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few will object to treating it in that language.
I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well.
The main advantage I see in geometric algebra is that its a unifying framework. The same langauge used to understand E&M in GA is also useful in quantum mechanics, general relativity, complex analysis, linear algebra, etc. I'd argue that physicists don't mind this mish-mash of notations because they've already put in the work to learn them all separtely but I think there are real advantages (especially pedagogical) to unifying them into one framework.
Re: Let's remove Quaternions from every 3D Engine
#57When 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.
But once you are in gimbal lock, what do you do?
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Re: Let's remove Quaternions from every 3D Engine
#58This article reads like 'I don't understand quaternions therefore they're bad.' Like what - they are the objectively best representation of rotations...
That's not at all what the author is saying. The author actually has a much deeper understanding of quaternions than most of their users and realizes that they are part of a much more general and useful structure known as the 3D geometric algebra and he advocates using that instead of quaternions.
GA(3) use objects that are isomorphic to quaternions to represent rotations but is much more comprehensible and generalizable than quaternions.
Re: Let's remove Quaternions from every 3D Engine
#59How do rotors map to euler angles?
The same way quaternions do.
Re: Let's remove Quaternions from every 3D Engine
#60Earlier quoted context omitted.
In my personal experience physicists are usually rather pragmatic about these things. Not many will support a "project of moving to geometric algebra" for its own sake, or the sake of perceived mathematical elegance - I know I wouldn't, even though I spent some time studying it just for fun. On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few…
> On the other hand, find a problem where geometric algebra gets you to a useful insight in a quicker or more natural way and few will object to treating it in that language. I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well. The main advanta…
I think that last point really needs to start with a thorough reform of basic mathematical education (starting from undergrad calculus) to have any chance of real success. The present mish-mash of notations, however disjointed, lends itself very well to teaching people who don't know much beyond basic real analysis and differential equations. You can't just throw them in a GA-based E&M class without completely changing the entire curriculum first.
Your first point got me curious, though. I found E&M in differential forms quite satisfying from an intellectual point of view (although of little practical benefit to the problems I was interested in compared to the classical approach, which I was plenty familiar with already). What advantages do you see for GA vs. differential forms in this context? This might make for a few interesting seminars with our math folks...