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Thanks for this. It always bugged me that quaternions didn't fit well with the linear algebra of computer graphics. You absolutely need quaternions, but every t

Thanks for this. It always bugged me that quaternions didn't fit well with the linear algebra of computer graphics. You absolutely need quaternions, but every time you do a rotation you have to convert to quaternions, do the rotation, then convert back. With geometric algebra (Clifford algebra), it's all just one system with the quaternions being a special case within that system. All of it makes more sense and computationally it's only slightly more work than linear algebra. Why don't we use Clifford algebra instead of linear algebra today? Because back in the 1800s there was something of a war between the people who liked Hamilton's formulation (quaternions, tensors, Clifford algebra) and those who liked vectors and matrices. Except in a few fields like relativity, the vector/matrix crowd won the war. And they probably shouldn't have.

bigbillheck 4y ago
> You absolutely need quaternions, You absolutely do not. Quaternions are (usually) the best solution, but they're not the only one. > And they probably shouldn't have Geometric algebra may or may not be better in certain low-dimensional applications like CG, but there's a whole lot more to linear algebra than that.
- jacobolus 4y ago
  Matrix algebra and geometric algebra are two different languages for many of the same topics/structures. If you like you can embed either one in the other (though embedding geometric algebra into matrix algebra is extremely cumbersome, sort of like rewriting your Python programs in assembly language). Geometric algebra has a much richer geometrical structure, but in some contexts the more basic structure of linear transformations is all you need, and matrices work just fine. In geometric algebra, you can extend a linear transformation of vectors by "outermorphism" to act on arbitrary multivectors.
CamperBob2 4y ago
I suspect that if people in the 1800s had been developing 3D computer graphics rather than electromagnetic theory, Hamilton's camp would have won the battle. But most people would probably say that vectors and matrices are a better fit for EM work.
- jacobolus 4y ago
  Clifford’s geometric algebra/calculus is a much better fit than the Gibbs/Heaviside concept of "vector" analysis for doing electrodynamics problems. Especially if you try to handle quantum mechanics and special relativity. Cross products, pseudovectors, etc. are tremendously confusing for students. Then throwing it all out the following school term in favor of cumbersome matrix representations makes things even worse. I suspect if Clifford had lived past ~35 years old we would have avoided a lot of the confusion of the 20th century.

bigbillheck 4y ago

> You absolutely need quaternions, You absolutely do not. Quaternions are (usually) the best solution, but they're not the only one. > And they probably shouldn't have Geometric algebra may or may not be better in certain low-dimensional applications like CG, but there's a whole lot more to linear algebra than that.

jacobolus 4y ago
Matrix algebra and geometric algebra are two different languages for many of the same topics/structures. If you like you can embed either one in the other (though embedding geometric algebra into matrix algebra is extremely cumbersome, sort of like rewriting your Python programs in assembly language). Geometric algebra has a much richer geometrical structure, but in some contexts the more basic structure of linear transformations is all you need, and matrices work just fine. In geometric algebra, you can extend a linear transformation of vectors by "outermorphism" to act on arbitrary multivectors.

CamperBob2 4y ago

I suspect that if people in the 1800s had been developing 3D computer graphics rather than electromagnetic theory, Hamilton's camp would have won the battle. But most people would probably say that vectors and matrices are a better fit for EM work.

jacobolus 4y ago
Clifford’s geometric algebra/calculus is a much better fit than the Gibbs/Heaviside concept of "vector" analysis for doing electrodynamics problems. Especially if you try to handle quantum mechanics and special relativity. Cross products, pseudovectors, etc. are tremendously confusing for students. Then throwing it all out the following school term in favor of cumbersome matrix representations makes things even worse. I suspect if Clifford had lived past ~35 years old we would have avoided a lot of the confusion of the 20th century.