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- URL: https://x.com/EricLengyel/status/1482866185490161667
- Author: Eric Lengyel (@EricLengyel)
- Posted: 2022-01-17 00:03:43
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1/ @EricLengyel
One of the biggest misunderstandings I see about quaternions is their dimensionality. Despite having 4 components, quaternions are three dimensional things that live and work in 3D space. 1/2
2/ @EricLengyel
• Quaternions arise as the geometric products of 3D vectors.
• These produce sums of scalars and 3D bivectors.
• That’s 4 numbers per quaternion, but it’s a combination of things that exist in the 3D geometric algebra.
• Each quaternion has an equivalent 3×3 matrix, not 4×4.
3/ @gottfired
@EricLengyel
That’s a bit too simplified. While normalized quaternions can be interpreted as rotation in Euclidean space. Quaternions are a 4 dimensional vector space. They are also a skew field which was the original interpretation of Hamilton as extension of real numbers.
4/ @EricLengyel
@gottfired
Yes, quaternions form an abstract 4D vector space, but that is not geometrically significant. Each of the components of a quaternion (and more generally, a 3D multivector) is one of 8 possible combinations of 3 physical dimensions. In that way, quaternions are undoubtedly 3D.
5/ @gottfired
@EricLengyel
I guess best comparison would be complex numbers which are a field. Normalized they are one dimensional representing 2d rotations. But that’s just a geometric interpretation. Complex numbers are useful in maths/physics for many other reasons.
6/ @gottfired
@EricLengyel
Same for quarternions. Geometry is just one subset and only for normalized quats which repr 3d rotations.
7/ @EricLengyel
@gottfired
Quaternions are homogeneous and do not need to be normalized, though it is often convenient to do so. In general, a quaternion of any magnitude transforms v with qvq⁻¹, but if q is unit length, then this simplifies to qvq̃ (where ~ is the reverse / conjugate).
8/ @gottfired
@EricLengyel
That’s exactly my point. If you explain complex numbers you’d never say that C only has 1 dof. Only as 2d rots this interpretation makes sense. But C is used almost exclusively for other stuff than representing rotations where a simple angle is good enough.
9/ @gottfired
@EricLengyel
But quats are useful outside of their convenience in 3d gfx as better Euler angles without gimbal lock. Same as C is very useful arguably way more so than the rotation interprets which afaik is hardly used.