6 ms·

Quaternions and spherical trigonometry

124 points by t55 2y ago

aap_ 2y ago
Interestingly showing spherical trig identities was how Hamilton demonstrated his quaternions to the royal irish academy when he found them. See (D) through (K): https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern1/ https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quater... When i first read this i found it very hard to understand, because i was unfamiliar with spherical trigonometry, but there's quite some beauty to be found there.
incognito124 2y ago
Related: https://eater.net/quaternions https://eater.net/quaternions
- t55 2y ago
  wow love the fact that you can interact with the videos
  - incognito124 2y ago
    That's what's most valuable to me too. If I remember correctly, Ben Eater (8bit CPU from scratch guy) and Grant Sanderson (3b1b guy) tried to do a little experiment on education with this guided-yet-interactive learning material back in 2018 (!!). As someone working in EDU sector of tech, I'm constantly amazed by what people come up with in order to explain things to other people (or even themselves). Whatever the experiment was, I'd say it was a success, and I hope we get to see more such material on other topics in future.
- FpUser 2y ago
  What a beautiful and high quality presentations. All the praises to the author.
- dang 2y ago
  Related: Visualizing quaternions (2018) - https://news.ycombinator.com/item?id=38043644 https://news.ycombinator.com/item?id=38043644 - Oct 2023 (42 comments) Visualizing quaternions: an explorable video series (2018) - https://news.ycombinator.com/item?id=31083042 https://news.ycombinator.com/item?id=31083042 - April 2022 (15 comments) Visualizing quaternions: An explorable video series - https://news.ycombinator.com/item?id=18310788 https://news.ycombinator.com/item?id=18310788 - Oct 2018 (32 comments)
- ge96 2y ago
  I still gotta grasp this for IMUs
  - Joel_Mckay 2y ago
    In general, the better sensor fusion algorithms incorporate kalman filtering. https://www.olliw.eu/2013/imu-data-fusing/#chapter23 https://www.olliw.eu/2013/imu-data-fusing/#chapter23 Best of luck =3
    - ge96 2y ago
      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
      - Joel_Mckay 2y ago
        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
- jdranczewski 2y ago
  This was an invaluable resource 5 years ago when I was working on a summer research project making a ray tracing-based optical levitation simulator - initially it felt a bit insane to try to deeply understand this obscure bit of maths to implement rotations, but once it clicked it clicked. Quaternions ended up being a super neat formalism for writing and computing rotational equations of motion. https://github.com/jdranczewski/optical-levitation-raytracing-experiments https://github.com/jdranczewski/optical-levitation-raytracin... for my repo, and https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 for the rotational dynamics with quaternions.
- deleted 2y ago
  [deleted]
- ks2048 2y ago
  FYI - if you're like me, you just looked at the page thinking it had some links to 3blue1brown videos. Not obvious that those are thumbnails link to interactive video apps. Very cool.
gamedever 2y ago
[flagged]
- t55 2y ago
  lmao i guess i'll be the last one scribbling i, j, k in my stone tablet
- seanhunter 2y ago
  Do you think Terence Tao doesn't understand Geometric Algebra? Just trying to understand where your hot take is coming from. My working hypothesis is he has at least an understanding of most fields of advanced maths and if he's looking into something it's because he thinks something interesting is there. That carries a lot of weight. Edit to add: he has done research into Quaternionic Hilbert spaces in the past https://pubs.aip.org/aip/jmp/article-abstract/37/11/5848/465974/On-the-structure-of-projective-group?redirectedFrom=fulltext https://pubs.aip.org/aip/jmp/article-abstract/37/11/5848/465...
  - Joel_Mckay 2y ago
    Quaternions have niche use-cases that greatly simplify some problems that otherwise create ambiguities using naive approaches. Many first encountered them in sensor fusion for IMU sensors, or 3D graphics. =3
- deleted 2y ago
  [deleted]
- deleted 2y ago
  [deleted]
andrewfromx 2y ago
like drawing triangles on a beach ball!
quantadev 2y ago
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 direction (perpendicular direction). What basic complex numbers represent is a way of doing rotations where something moves from one direction towards it's orthogonal. That's what Euler's Formula is about also, which shows the relationship of 'e' and 'i' in this of course. Now what Quaternions represents is the realization that if complex numbers have two components (real, imaginary) then we can treat each of those as a base vector and find a sort of 'next level up' orthogonality to each one individually. I'm not good enough at math/geometry to know if this kind of 'next level up' bifurcation of dimensionality extends up past Quaternions or not (like something called Octernions, 16ions, 32ions, 64ions, etc), but it seems like is would?
- hgomersall 2y ago
  Geometric algebra would be what you're looking for here. This is a great intro to the topic: https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-ar...
  - AnIrishDuck 2y ago
    Also the (provocatively titled) "Let's Remove Quaternions from every 3d Engine" [1] Spoiler alert: rotors are mechanically identical to quaternions, while being easier to understand. If you understand rotors, you understand quaternions. You can fit the laws you need to understand rotors on a business card. Plus, rotors abstract to higher and lower (well, there's only one plane and its two respective orientations in 2d, but still) dimensions. Complex numbers as planes (bivectors in GA parlance) has been the most mind-opening mathematical concept I've been exposed to in the last decade. The associated geometric product has helped me better understand concepts (like "handedness") that troubled me during undergrad engineering. 1. https://marctenbosch.com/quaternions/ https://marctenbosch.com/quaternions/
alfiedotwtf 2y ago
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 Algebra isn’t it? Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?! Please chime in if you know what I’m talking about and can confirm this or shoot it down.
- particleguy 2y ago
  Yes, Quantum Field Theory can be explained through Lie groups. SU(2) is isomorphic to the quaternions of norm 1, and SU(2) is important if you want to understand the Lorentz group and Poincare group, which represent the symmetries of spacetime and special relativity. Check out the text book Physics From Symmetry by Jakob Schwichtenberg if you would like an approach that derives modern physics primarily from algebra
  - alfiedotwtf 2y ago
    OMG thank you. Purchasing right now! You DO NOT understand how happy I am right now. Truely! I did general physics for a year at uni as part of my Computer Engineering course, then switching to Computer Science where I picked up a year of quantum mechanics. Since then whenever I lay in bed and thought about physics I would end up awake for hours. So damn interesting but the maths always held me back, so sadly gave up. I don’t know what’s changed (maybe maturity or maybe Vyvanse lol) but I’m slowly putting the pieces together. It’s always been in my outer periphery but still out of reach. Your confirmation has and will change my life. Maybe not career wise or life altering seen from the outside, but hot damn you have at least cleared my constant nagging guilt for not perusing maths and physics because you’ve just made it slightly closer within reach. Can’t wait for the book to arrive. Thank you!!!
    - raphlinus 2y ago
pjbk 2y ago
You can certainly use quaternions (as Hamilton demonstrated) or a Clifford algebra to recover the spherical trigonometry laws, but plain vectors work too in a short derivation. It is actually one of the simple exercises introducing reciprocal bases in Louis Brand's book Vector and Tensor Analysis (https://archive.org/details/vectortensoranal00branrich https://archive.org/details/vectortensoranal00branrich) or its abridged version, Vector Calculus.

124 points by t55 2y ago

aap_ 2y ago

Interestingly showing spherical trig identities was how Hamilton demonstrated his quaternions to the royal irish academy when he found them. See (D) through (K): https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern1/ https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quater... When i first read this i found it very hard to understand, because i was unfamiliar with spherical trigonometry, but there's quite some beauty to be found there.

incognito124 2y ago

Related: https://eater.net/quaternions https://eater.net/quaternions

t55 2y ago
wow love the fact that you can interact with the videos
- incognito124 2y ago
  That's what's most valuable to me too. If I remember correctly, Ben Eater (8bit CPU from scratch guy) and Grant Sanderson (3b1b guy) tried to do a little experiment on education with this guided-yet-interactive learning material back in 2018 (!!). As someone working in EDU sector of tech, I'm constantly amazed by what people come up with in order to explain things to other people (or even themselves). Whatever the experiment was, I'd say it was a success, and I hope we get to see more such material on other topics in future.
FpUser 2y ago
What a beautiful and high quality presentations. All the praises to the author.
dang 2y ago
Related: Visualizing quaternions (2018) - https://news.ycombinator.com/item?id=38043644 https://news.ycombinator.com/item?id=38043644 - Oct 2023 (42 comments) Visualizing quaternions: an explorable video series (2018) - https://news.ycombinator.com/item?id=31083042 https://news.ycombinator.com/item?id=31083042 - April 2022 (15 comments) Visualizing quaternions: An explorable video series - https://news.ycombinator.com/item?id=18310788 https://news.ycombinator.com/item?id=18310788 - Oct 2018 (32 comments)
ge96 2y ago
I still gotta grasp this for IMUs
- Joel_Mckay 2y ago
  In general, the better sensor fusion algorithms incorporate kalman filtering. https://www.olliw.eu/2013/imu-data-fusing/#chapter23 https://www.olliw.eu/2013/imu-data-fusing/#chapter23 Best of luck =3
  - ge96 2y ago
    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
    - Joel_Mckay 2y ago
      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
jdranczewski 2y ago
This was an invaluable resource 5 years ago when I was working on a summer research project making a ray tracing-based optical levitation simulator - initially it felt a bit insane to try to deeply understand this obscure bit of maths to implement rotations, but once it clicked it clicked. Quaternions ended up being a super neat formalism for writing and computing rotational equations of motion. https://github.com/jdranczewski/optical-levitation-raytracing-experiments https://github.com/jdranczewski/optical-levitation-raytracin... for my repo, and https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 for the rotational dynamics with quaternions.
deleted 2y ago
[deleted]
ks2048 2y ago
FYI - if you're like me, you just looked at the page thinking it had some links to 3blue1brown videos. Not obvious that those are thumbnails link to interactive video apps. Very cool.

gamedever 2y ago

[flagged]

t55 2y ago
lmao i guess i'll be the last one scribbling i, j, k in my stone tablet
seanhunter 2y ago
Do you think Terence Tao doesn't understand Geometric Algebra? Just trying to understand where your hot take is coming from. My working hypothesis is he has at least an understanding of most fields of advanced maths and if he's looking into something it's because he thinks something interesting is there. That carries a lot of weight. Edit to add: he has done research into Quaternionic Hilbert spaces in the past https://pubs.aip.org/aip/jmp/article-abstract/37/11/5848/465974/On-the-structure-of-projective-group?redirectedFrom=fulltext https://pubs.aip.org/aip/jmp/article-abstract/37/11/5848/465...
- Joel_Mckay 2y ago
  Quaternions have niche use-cases that greatly simplify some problems that otherwise create ambiguities using naive approaches. Many first encountered them in sensor fusion for IMU sensors, or 3D graphics. =3
deleted 2y ago
[deleted]
deleted 2y ago
[deleted]

andrewfromx 2y ago

like drawing triangles on a beach ball!

quantadev 2y ago

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 direction (perpendicular direction). What basic complex numbers represent is a way of doing rotations where something moves from one direction towards it's orthogonal. That's what Euler's Formula is about also, which shows the relationship of 'e' and 'i' in this of course. Now what Quaternions represents is the realization that if complex numbers have two components (real, imaginary) then we can treat each of those as a base vector and find a sort of 'next level up' orthogonality to each one individually. I'm not good enough at math/geometry to know if this kind of 'next level up' bifurcation of dimensionality extends up past Quaternions or not (like something called Octernions, 16ions, 32ions, 64ions, etc), but it seems like is would?

hgomersall 2y ago
Geometric algebra would be what you're looking for here. This is a great intro to the topic: https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-ar...
- AnIrishDuck 2y ago
  Also the (provocatively titled) "Let's Remove Quaternions from every 3d Engine" [1] Spoiler alert: rotors are mechanically identical to quaternions, while being easier to understand. If you understand rotors, you understand quaternions. You can fit the laws you need to understand rotors on a business card. Plus, rotors abstract to higher and lower (well, there's only one plane and its two respective orientations in 2d, but still) dimensions. Complex numbers as planes (bivectors in GA parlance) has been the most mind-opening mathematical concept I've been exposed to in the last decade. The associated geometric product has helped me better understand concepts (like "handedness") that troubled me during undergrad engineering. 1. https://marctenbosch.com/quaternions/ https://marctenbosch.com/quaternions/

alfiedotwtf 2y ago

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 Algebra isn’t it? Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?! Please chime in if you know what I’m talking about and can confirm this or shoot it down.

particleguy 2y ago
Yes, Quantum Field Theory can be explained through Lie groups. SU(2) is isomorphic to the quaternions of norm 1, and SU(2) is important if you want to understand the Lorentz group and Poincare group, which represent the symmetries of spacetime and special relativity. Check out the text book Physics From Symmetry by Jakob Schwichtenberg if you would like an approach that derives modern physics primarily from algebra
- alfiedotwtf 2y ago
  OMG thank you. Purchasing right now! You DO NOT understand how happy I am right now. Truely! I did general physics for a year at uni as part of my Computer Engineering course, then switching to Computer Science where I picked up a year of quantum mechanics. Since then whenever I lay in bed and thought about physics I would end up awake for hours. So damn interesting but the maths always held me back, so sadly gave up. I don’t know what’s changed (maybe maturity or maybe Vyvanse lol) but I’m slowly putting the pieces together. It’s always been in my outer periphery but still out of reach. Your confirmation has and will change my life. Maybe not career wise or life altering seen from the outside, but hot damn you have at least cleared my constant nagging guilt for not perusing maths and physics because you’ve just made it slightly closer within reach. Can’t wait for the book to arrive. Thank you!!!
  - raphlinus 2y ago

pjbk 2y ago

You can certainly use quaternions (as Hamilton demonstrated) or a Clifford algebra to recover the spherical trigonometry laws, but plain vectors work too in a short derivation. It is actually one of the simple exercises introducing reciprocal bases in Louis Brand's book Vector and Tensor Analysis (https://archive.org/details/vectortensoranal00branrich https://archive.org/details/vectortensoranal00branrich) or its abridged version, Vector Calculus.