Connecting GA Rotation Formulas
Вставка
- Опубліковано 30 вер 2023
- In this short, I show how the n-dimensional rotation formula in geometric algebra is related to the idea that rotations can be calculated by decomposing the vector into parts that are parallel and perpendicular to the plane of rotation.
Discord: / discord
Patreon: / sudgylacmoe
Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb
I never understood why quaternions required a sandwich product when complex numbers (a subspace) don't, until I realized that this also applies to quaternions if everything is contained within one plane.
You don't need to cancel the perpendicular part if there is no perpendicular part!
Yeah I often utilize this to make my rotations simpler. Even in a 4D algebra like 3D PGA, as long as you are in a two-dimensional subspace you can use the simple rotation formula.
Thanks this is awesome
woooooooow
what a brilliant explanation!
I guess, this way you can proove rotation formula for any dimention by induction
You don’t even need induction, because this works for any number of dimensions immediately! The only difference is the dimension of the “perpendicular part”, which was never important to this derivation.
@@jakobr_ that's actually what induction is - you prove n+1 from n (to put it simple)
@@dzuchun Yeah, I know what induction is, and I’m saying this isn’t it. Nothing about this relates to stepping up from one case to the next.
Take the triangular numbers formula, for example. You could prove T_n = n*(n+1)/2 for all numbers at once by using the geometry of squares arranged in a staircase pattern, equivalent to algebraically pairing every number in the sum to its descending partner, or you could use induction and say that T_1 = 1*2/2 = 1 and also T_(n-1) = n*(n-1)/2 implies T_n = (n^2-n)/2 + 2n/2 = (n^2+n)/2 = n*(n+1)/2.
The first method proves the formula for all numbers at once, there’s no cascade of implication like there is in induction. That’s what I’m saying this video is related to.
@@jakobr_ oh wow. My bad, I see now
the intuition of parallel part being one-dimentional really confused me. In fact, it's a perpendicular part that's guaranteed to be two-dimentional. I guess, I'm too 3d-oriented even for 2d
thanks again
@@dzuchun It’s all good! Yeah it’s super weird to think about parallel and perpendicular parts of spaces outside of the context of 3D.
Like, if you think about a rotation in 2D space, the “perpendicular” part, the “axis”, would be 2-2=0D, a point. And in 4D, the “axis” would be yet another plane! It doesn’t even make sense to use a word like “axis” outside of 3D. Parallel and perpendicular are just big words that mean “this is the stuff that changes with the rotation totally” and “this is the stuff that doesn’t change at all”
Nice!
Thanks for this!
Oh, it's actually starting to click for me!
Cool stuff
I love sandwich products. They make my head spin.5
Uhha uhha ya i get it numbers
Numbers named after letters
Dang