5 ms·
Since there's currently an article about quaternions on the front page, it felt like a good time to bring up bivectors, which in my opinion are a much more eleg
by kilovoltaire 5y ago
Since there's currently an article about quaternions on the front page, it felt like a good time to bring up bivectors, which in my opinion are a much more elegant way to do rotations.
Particularly, I think this part is insightful:
> In 3D, a bivector has three coordinates, one per plane: (xy, xz, and yz). Vectors also have three coordinates, one per axis (x, y and z). Each plane is perpendicular to one axis. This is a coincidence that only happens in three dimensions and it is why historically we have been confusing bivectors with vectors.
- zardo 5y agoIt's also nice that doing 2D physics, you can represent torque without invoking an extra orthogonal spatial dimension.
- idealmedtech 5y ago> Each plane is perpendicular to one axis. This is a coincidence that only happens in three dimensions This part is confusing to me; in any number of dimensions, your orthonormal basis vectors (coordinate vectors) will by definition have tangent vector spaces (planes). There's nothing special about 3 in this regard, unless I'm misunderstanding something.
- Sukera 5y agoI think they're saying the planes in particular are the tangent vector space in 3 dimensions. They're no longer the tangent vector space in 2,4,5.. dimensions, but they're still a thing there as well.
- krastanov 5y agoIn N dimensions you do indeed have N hyperplanes of dimension N-1. But you have N*(N-1)/2 planes of dimension 2. Then one needs to explain why is rotation in general related to a 2D plane, not to an axis. Tautologically, it is so because rotation is when you oscillate between two directions (the two vectors of the 2D plane). It is not "rotating around an axis", because there is no consistent way to give one single angle of rotation around an axis if you are in 4+ dimensions.