3 ms·

I’ve been exploring this idea as well however I have a hunch it’s not actually octonions but dual quaternions as they are the perfect formalism for representing

I’ve been exploring this idea as well however I have a hunch it’s not actually octonions but dual quaternions as they are the perfect formalism for representing 3d movement over time. And to add to that, the are a Lie group I.e. they are anticommutative I.e. AB=-BA. I’ve also been exploring this relationship between dual quaternions and linear logic. It’s pretty wild. I’m curious if anyone has any opinions on this.

stochastic_monk 8y ago
Do you have an understanding of Clifford algebras? I’ve read a little about them here on hn, and they seem quite powerful. I don’t understand them enough to know if they could also be an appropriate abstraction.
- adamnemecek 8y ago
  Yes! Dual quaternions are a Cl(2,0,1) Clifford algebra. Good ol Cliffy invented them too. http://www.chinedufn.com/dual-quaternion-shader-explained/ http://www.chinedufn.com/dual-quaternion-shader-explained/ Check the guy on the right getting deformed like an idiot. Compare it with the smooth criminal on the right.
  - electricslpnsld 8y ago
    > http://www.chinedufn.com/dual-quaternion-shader-explained/ http://www.chinedufn.com/dual-quaternion-shader-explained/ Offset curve deformations blow dual quaternion's out of the water for rigging!
    - adamnemecek 8y ago
      Tell me more. I briefly looked it up and I do agree it does look somewhat nicer. However how's the performance.
      - electricslpnsld 8y ago
        Fast enough for use in live character rigs in film running on the CPU -- I know the technique is used at Sony Imageworks and at Dreamworks. I can't comment on whether the method is suitable for live rigs in games, or how it would translate the GPU.
- joppy 8y ago
  There is a process where you put in a space (number of dimensions) and a metric (Euclidean, for example) and produce a Clifford algebra. This is also sometimes called “geometric algebra”, although there are a lot of sensationalist posts using this name on the internet. This Clifford algebra has an odd and even part, and the even part is an algebra in its own right, and is very useful for representing “rotations” in the space. With the Euclidean metric, with one input dimension, the even Clifford algebra is just the reals. With two dimensions, you get the complex numbers. With three, the quaternions. However, since a Clifford algebra is always associative, you don’t get the octonions in this way. Finally, if you put in four dimensions and the minkowski metric, the even Clifford algebra construction gives you a lovely algebra representing spatial rotations in three of the dimensions, and “boosts” along the time direction - exactly what you need to do calculations in special relativity.
  - eggy 8y ago
    I am working my way through Understanding Geometric Algebra by Kenichi Kanatani up to the Grassmann Algebra chapter, so I can then pick up John Browne's Grassmann Algebra Volume 1 Foundations. Kenichi's book works from basic vector calculus, Hamilton, Grassmann and Clifford. Grassmann was not even a mathematician by trade, but a school teacher and a linguist. He translated the Rig Veda. His big mathematical work did not get the attention when written as it did posthumously. I know things with cool names like differntial forms and fiber bundle space [1]. I'm becoming quite the Grassmann fanboy! I am thinking if I can learn these algebras (Grassmann, Clifford, etc...) I will be able to deal with more abstract thinking in geometry and handle it more concisely and logically. Almost like when I use J programming language for math! [2] I think Quarternions don't hold after 3 dimensions, so what about Octonions? [1] https://galileospendulum.org/2012/04/26/everything-is-geometrical-hermann-grassmanns-algebra/ [2] jsoftware.com
ssivark 8y ago
Incidentally, I've also been thinking about dual quaternions for modeling rigid objects in 3d. For anyone who's curious about the topic, these seem to be a decent set of notes: http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pdf http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pd... Linear logic always reminded me of the no-cloning property in quantum physics, where a resource can only be used once -- no copying for multiple use. Can you elaborate on the sense in which you think of linear logic as "squaring to zero"? UPDATE: A little googling throws up this: http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/msg00520.html http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/...
- adamnemecek 8y ago
  Good point about no copy. So this is kinda roundabout however there’s homotopy type theory which states that topology=logic=type theory. For a second let's say that theres a fourth "element" in this equation and that's quaternions (this isn't a stretch I can provide some papers that hint at this). Linear logic is classical logic with the added reference to time without explicitly referencing time. Dual quaternions all about modeling deformation over time. Therefore, I would conclude that linear logic and dual quaternions are related.
chombier 8y ago
one of the most intriguing/funny things about dual numbers (and dual quaternions in particular) is how they encode the derivative of the real part into the dual part, and how everything plays out nicely algebraically. this could be a nice implementation trick for an automatic differentiation framework.

stochastic_monk 8y ago

Do you have an understanding of Clifford algebras? I’ve read a little about them here on hn, and they seem quite powerful. I don’t understand them enough to know if they could also be an appropriate abstraction.

adamnemecek 8y ago
Yes! Dual quaternions are a Cl(2,0,1) Clifford algebra. Good ol Cliffy invented them too. http://www.chinedufn.com/dual-quaternion-shader-explained/ http://www.chinedufn.com/dual-quaternion-shader-explained/ Check the guy on the right getting deformed like an idiot. Compare it with the smooth criminal on the right.
- electricslpnsld 8y ago
  > http://www.chinedufn.com/dual-quaternion-shader-explained/ http://www.chinedufn.com/dual-quaternion-shader-explained/ Offset curve deformations blow dual quaternion's out of the water for rigging!
  - adamnemecek 8y ago
    Tell me more. I briefly looked it up and I do agree it does look somewhat nicer. However how's the performance.
    - electricslpnsld 8y ago
      Fast enough for use in live character rigs in film running on the CPU -- I know the technique is used at Sony Imageworks and at Dreamworks. I can't comment on whether the method is suitable for live rigs in games, or how it would translate the GPU.
joppy 8y ago
There is a process where you put in a space (number of dimensions) and a metric (Euclidean, for example) and produce a Clifford algebra. This is also sometimes called “geometric algebra”, although there are a lot of sensationalist posts using this name on the internet. This Clifford algebra has an odd and even part, and the even part is an algebra in its own right, and is very useful for representing “rotations” in the space. With the Euclidean metric, with one input dimension, the even Clifford algebra is just the reals. With two dimensions, you get the complex numbers. With three, the quaternions. However, since a Clifford algebra is always associative, you don’t get the octonions in this way. Finally, if you put in four dimensions and the minkowski metric, the even Clifford algebra construction gives you a lovely algebra representing spatial rotations in three of the dimensions, and “boosts” along the time direction - exactly what you need to do calculations in special relativity.
- eggy 8y ago
  I am working my way through Understanding Geometric Algebra by Kenichi Kanatani up to the Grassmann Algebra chapter, so I can then pick up John Browne's Grassmann Algebra Volume 1 Foundations. Kenichi's book works from basic vector calculus, Hamilton, Grassmann and Clifford. Grassmann was not even a mathematician by trade, but a school teacher and a linguist. He translated the Rig Veda. His big mathematical work did not get the attention when written as it did posthumously. I know things with cool names like differntial forms and fiber bundle space [1]. I'm becoming quite the Grassmann fanboy! I am thinking if I can learn these algebras (Grassmann, Clifford, etc...) I will be able to deal with more abstract thinking in geometry and handle it more concisely and logically. Almost like when I use J programming language for math! [2] I think Quarternions don't hold after 3 dimensions, so what about Octonions? [1] https://galileospendulum.org/2012/04/26/everything-is-geometrical-hermann-grassmanns-algebra/ [2] jsoftware.com

ssivark 8y ago

Incidentally, I've also been thinking about dual quaternions for modeling rigid objects in 3d. For anyone who's curious about the topic, these seem to be a decent set of notes: http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pdf http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pd... Linear logic always reminded me of the no-cloning property in quantum physics, where a resource can only be used once -- no copying for multiple use. Can you elaborate on the sense in which you think of linear logic as "squaring to zero"? UPDATE: A little googling throws up this: http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/msg00520.html http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/...

adamnemecek 8y ago
Good point about no copy. So this is kinda roundabout however there’s homotopy type theory which states that topology=logic=type theory. For a second let's say that theres a fourth "element" in this equation and that's quaternions (this isn't a stretch I can provide some papers that hint at this). Linear logic is classical logic with the added reference to time without explicitly referencing time. Dual quaternions all about modeling deformation over time. Therefore, I would conclude that linear logic and dual quaternions are related.

chombier 8y ago

one of the most intriguing/funny things about dual numbers (and dual quaternions in particular) is how they encode the derivative of the real part into the dual part, and how everything plays out nicely algebraically. this could be a nice implementation trick for an automatic differentiation framework.