AHHHHHH I finally understand quaternions!! XDD :DD
I always thought they were neat fun thing in abstract algebra and a neat but arbitrary way of representing rotations with unnecessarily extra numbers.
If someone had just said to me this sentence I would have seen how insanely perfect they are!! :D
As rotations, the i/j/k coefficients are the x/y/z components of the axis-of-rotation vector, and the fourth lone number is the amount of the rotation in radians!
🇳🇱
@codepuppy Way to go, its great to learn a new thing...
Here is the catch to really understanding Quaternions and how to use them though... do you understand what a quaternion can accomplish and why one might use it over the simpler to understand and more generally useful vector math? Afterall if you use vectors you can work in any number of dimensions whereas quaternions are limited 3D rotations or 4D points. So understanding why quaternions are superior for specific situations is really the key to getting them and using them.
@freemo Ahh I see, like how the cross product is dimensionally-limited :3
Well I'm mostly focused on writing 3D computational geometry code right now, and mostly I'm just excited that I already basically understand this thing everyone's talking about XD
I thought quaternions had some insanely arbitrary arcane set of formulas for transforming the x/y/z of a 3D vector into the rotated vector and I should learn them someday when I had time to spare (haha) but they weren't essential.
@codepuppy they are just complex numbers taken out to 4D instead of 2D, you can extend this to even higher dimensions with things like octonians and sedenions. Each level looses something though for example quaternions loose commutativity. but otherwise they arent that obscure when you think about it.
@codepuppy @freemo the mathematical answer is "Hurwitz's theorem" (https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)), but I tend to go with Hamilton's original take that stuff just doesn't work out in 3D, and you need a fourth dimension for things to fit into place.
@codepuppy @freemo and the follow up to Hurwitz is the Cayley-Dickson construction (https://en.wikipedia.org/wiki/Cayley–Dickson_construction) which is, crudely put, a way to build complex numbers from reals, quaternions from complex numbers, and so forth.
@freemo well they did seem obscure to me in that 2D complex numbers come from negative square roots and logarithms,
whereas quaternions just seemed arbitrary and "made up"—like that the multiplication table could have been different and would have been a different extension of complex numbers to 4D!
is that not true? is there something about it that makes it a unique extension of complex numbers in 4D? :>