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Quaternions and spherical trigonometry

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Re: Quaternions and spherical trigonometry

#31

Post by quantadev »

Sniffnoy wrote: post #22
quantadev wrote: post #16

Just a tiny rant: In my view complex numbers are really about the concept of "orthogonality". The complex 'dimension' is orthogonal to the 'real' dimension, but anything in reality that's a continuum of values can be seen as a dimension, and therefore each one must have an orthogonal. That is, whenever you have a direction in a higher dimensional space (regardless of dimensionality) any vector will have a normal dire…

Octonions and so on up are indeed a thing, but I don't think they do what you want. Even aside from the fact that they're restricted to power-of-2 dimensions, their algebraic properties get worse as you iterate the Cayley-Dickson process. The octonions aren't even an associative algebra, although they do have some weaker associativity properties which I'll skip detailing here. The quaternions are as far as most mathe…

From your post I can tell you're way better at math/geometry than me, but I understood 80% of that. :)

Cayley-Dickson is interesting especially for Physics of course, because it brings in the concept of 'variable dimensions'. I think the flattening of objects, and the stopping of clocks (in Relativity), due to Lorentz effects in Minkowski space both on Black Hole Event Horizons and for objects approaching light speed (anywhere Lorentz holds) is, at the limits, ultimately the loss of a dimension, which would be my overall interpretation of what Cayley-Dickson is about too, in very broad terms.

So if Minkowski space is 4 dimensional, there would be some geometry for a 5-Dim Minkowski and it would use Octonians maybe, and that would be the geometry of the universe our universe is "embedded in"...I mean assuming of course you believe our universe is a Black Hole and we are all on an Event Horizon embedded in a 5D universe. Ya know, as one does. lol.

Re: Quaternions and spherical trigonometry

#32

Post by quantadev »

nh23423fefe wrote: post #25
quantadev wrote: post #16

Just a tiny rant: In my view complex numbers are really about the concept of "orthogonality". The complex 'dimension' is orthogonal to the 'real' dimension, but anything in reality that's a continuum of values can be seen as a dimension, and therefore each one must have an orthogonal. That is, whenever you have a direction in a higher dimensional space (regardless of dimensionality) any vector will have a normal dire…

I don't agree. Complex numbers are the algebraic closure of the reals. Or the quotient of the real polynomial ring by (x^2+1=0). Or whatever other construction. The multiplication rule is the essence of C. Orthogonality is captured linear algebra over R^2, but R^2 isn't a field or an algebra.

Sniffy nailed it (above) when he mentioned Cayley-Dickson. That's exactly what I was getting at, and had never researched.

Re: Quaternions and spherical trigonometry

#34

Post by alfiedotwtf »

aap_ wrote: post #29
alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as havi…

Woah. Ok, that model totally blew my mind. I never understood what angular momentum had anything to do with it, but the gyroscope analogy makes total sense. Thank you!

Re: Quaternions and spherical trigonometry

#36

Post by chadcmulligan »

aap_ wrote: post #29
alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as havi…

Sounds interesting - would you have a good intro book on spinors you could recommend?

Re: Quaternions and spherical trigonometry

#37

Post by ge96 »

Joel_Mckay wrote: post #35
ge96 wrote: post #15

Earlier quoted context omitted.

I still gotta grasp this for IMUs

In general, the better sensor fusion algorithms incorporate kalman filtering. https://www.olliw.eu/2013/imu-data-fusing/#chapter23 Best of luck =3

Yeah something else I need to do. lazy question, if I have an IMU that is swaying around as I try to move in a linear direction (e.g. Forward) is that something this kind of filtering would be used for? Regarding displacement estimation.

Edit: I get fusion is regarding multiple sensors

Re: Quaternions and spherical trigonometry

#38

Post by JadeNB »

alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

> Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?!

That's a great way to understand them if you are already comfortable with the algebra of spinors and spin groups, but it doesn't short-circuit the history—spinors were so called after quantum spin (https://en.wikipedia.org/wiki/Spinor#History), and I believe that was so called because, yes, it was envisioned as something at least conceptually spinning.

Re: Quaternions and spherical trigonometry

#39

Post by jeffwass »

alfiedotwtf wrote: post #24

Hold up! Omg, can someone who’s done physics chime in please… whenever I’ve looked at GUT etc, I’ve always seen U(n), SU(n), but never knew what they were - are they what’s referred to in this article? Is that just the Unitary Group and Special Unitary Group??! All that time I thought it was all impenetrable but it’s just algebra? Omg wow... the theoretical physics I’m talking about is just quaternions and Lie Algebr…

Quantum spin is an intrinsic angular momentum of a particle. It's angular momentum that is 'just there' as a key component of the particle.

In early days it was hypothesised that particles were spinning about their own axes, but this isn't accurate.

All the interesting stuff of Spin from its quantizable nature, the non-commutatability of spin measurements along orthogonal directions, the very different fundamental behavior of particles with half-integer spin (Fermions, eg Electrons, Protons) vs integer spin (Bosons eg Photons), how Spins interact (eg spins of say two electrons with half-integer spin interacting as a Spin-0 Boson in a Cooper Pair of a superconductor), or spin interacting with orbital angular momentum eg electron spin interacting with it's orbit around proton in an atom.

At the end of the day Spin isn't a terrible name for it.

I believe Spinor vectors are merely named after the eigenvectors used to represent spin itself, not the other way around as you suggested.

Re: Quaternions and spherical trigonometry

#40

Post by Joel_Mckay »

ge96 wrote: post #37
Joel_Mckay wrote: post #35

Earlier quoted context omitted.

In general, the better sensor fusion algorithms incorporate kalman filtering. https://www.olliw.eu/2013/imu-data-fusing/#chapter23 Best of luck =3

Yeah something else I need to do. lazy question, if I have an IMU that is swaying around as I try to move in a linear direction (e.g. Forward) is that something this kind of filtering would be used for? Regarding displacement estimation. Edit: I get fusion is regarding multiple sensors

Normally, there are several types of sensors available, but most use 9DOF packages (3-axis gyroscope, accelerometer, and magnetometer)

gyroscope: fast over-sampled low-pass filter, but slowly drifts compounding heading errors

accelerometer: relatively stable, but dead-reckoning errors compound quickly

magnetometer: best stability, but low-sample rate and vulnerable to metal/magnets fooling/blinding the sensors

The fusion algorithms usually weights which data is consistent with the motion path, and attenuates the estimated pose errors.

Notably, not all sensors are equal quality, but there are probably better options now. =3

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