Thread 16752573

You didn't fall for the vector psyop did you, anon?

what is that even supposed to mean?

>>16752576
one can basically have quaternions through vectors, but OP prefers the algebra, i think

>>16752573 (OP)
What psyop? Rotations of 3-vectors form their own group, SO(3). It is generated by 3 skew-symmetric linearly-independent 3x3 matrices that one exponentiates. They don’t commute, making it a big mess.

There are three ways out of this. Euler angles are the first way, but it’s dogshit. The group of Euler rotations is SO(1)^3, which is homeomorphic to a 3-torus, not S^3/(Z/2) = SO(3). It’s only locally homemorphic to SO(3). That’s why you get singularities like gimbal lock and undefined longitude of the ascending node at zero inclination. It may be intuitive, but it’s fundamentally flawed and nasty to work with.

Then there’s quaternions. There is an exceptional isomorphism of Lie algebras so(3) = sp(1), which leads to a Lie group isomorphism SO(3) = Sp(1)/(Z/2). This is a proper homeomorphism with no singular behavior. The only issue is that you have to take account of the redundant plus-minus signs (a rotation by q is the same as a rotation by -q).

The third way is the exceptional isomorphism so(3) = sl(2), which leads to the Lie group homomorphism SO(3) = SU(2)/(Z/2). You’re now working with unitary 2x2 complex matrices. This is more computationally expensive than quaternions, but you’re now working with matrices, which are easier to handle and understand. This is the way physicists represent rotations. It becomes important in quantum mechanics where one takes projective representations into account to work with half-integer spin.

>>16752573 (OP)
A value can have a direction AND a magnitude?
Ya nah. Miss me w/ dat shit.

>>16752619
vector analysis is just really easy to grok, even with all the inherent limitations and nuances when describing things that really aren't best described that way.

>>16752573 (OP)
Nearly, they had my mind crossed until i drove a wedge through its ambiguity between polar and axial vectors

>>16752851
>that really aren't best described that way
skill issue

>>16752573 (OP)
wait until you have to deal with tensors and graphs

>>16752573 (OP)
you never was judged by your basis vectors

>>16752619
I don't know which is better, 2x2 complex matrix vs 3x3 real matrix
https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Conversion_to_and_from_the_matrix_representation

>>16752573 (OP)
Math this complex is pushing the limits of what ordinary states of waking consciousness can simulate.

>>16752619
>They don’t commute, making it a big mess.
>There are three ways out of this.
3D rotations are non-commutative in general. Doing successive rotations with quaternions is no more commutative than with rotation matrices (i.e. multiplication of quaternions is not commutative)

>>16755491
but muh my game engine needs interpolation

>>16755906
the one compelling reason to use quaternions that i'm aware of, and its a good one
lots of people will point out gimbal lock as an issue, but don't realize it's an issue only when you're tracing a path in rotation space, e.g. gimbal lock isn't an issue when representing finite rotations as matrices, no matter how they get parameterized.

>>16755906
If you just want to interpolate between two configurations, you don't need to treat the whole transformation as a set of consecutive rotations. You calculate the end configuration as a single step, and then calculate desired number of interpolation points separately

>>16755906
You only need to precalculate nine values and you can smoothy rotate around an axis defined by a unit vector. It only needs nine multiplications and six additions.