3 ms·

Quaternions, Interpolation and Animation (1998) [pdf]

63 points by teleforce 3y ago

iTokio 3y ago
1998
tbm57 3y ago
These were considered to be satanic black magic when they were first discovered. this is the 'quaternomicon'
- teleforce 3y ago
  Oliver Heaviside and Lord Kelvin both considered it as an evil concept and strongly believed that Maxwell was misled by using it to derive Maxwell's Equations. Heaviside then come up with the famous modern version of Maxwell's Equations with four equations but now ironically all of them can be represented with one concise single equation using quaternion based on geometric algebra [1]. It's probably worth to mention that Einstein was using quaternion concept when discovering his infamous special relativity [2]. It's interesting to note that complex number was once had the same fate as quaternion and it was also considered as an evil concept at the time by many well-known mathematicians not unlike the concept of zero/sifar/cipher when it's first invented and then popularized by the Arab mathematicians. Modern engineering achievements for example the wonderful world of mobile and digital communication are all based on complex number concept namely in-phase and quadrature or I/Q for short. This venerable concept of I/Q however is limited once you wan to include and consider polarization of the EM wave and this is when quaternion become very handy and useful [3]. I'd not be surprised that the next generation of wireless mobile i.e. 6G will be based on this new 3D quaternion based I/Q or I/Q/P (not yet a concept maybe soon), similar to the impact of I/Q as the basis of successful 2G to the 5G with their OFDM. [1] A derivation of the quaternion Maxwell’s equations using geometric algebra: http://peeterjoot.com/2018/03/05/a-derivation-of-the-quaternion-maxwells-equations-using-geometric-algebra/ http://peeterjoot.com/2018/03/05/a-derivation-of-the-quatern... [2] Quaternions in University-Level Physics Considering Special Relativity: https://arxiv.org/ftp/physics/papers/0308/0308017.pdf https://arxiv.org/ftp/physics/papers/0308/0308017.pdf [3] Modeling dual polarization wireless fading channels using quaternions: https://ieeexplore.ieee.org/document/1708024 https://ieeexplore.ieee.org/document/1708024
sxp 3y ago
It's somewhat silly to read about animation using a static doc. My preferred intro to quaternions in the context of motion is the pair of articles at https://acko.net/tag/quaternions/ https://acko.net/tag/quaternions/. They were critical helping me learn quaternions for computer graphics. I also recommend https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ as a primer if you aren't comfortable with regular complex numbers.
- Culonavirus 3y ago
  That site is pretty amazing.
j2kun 3y ago
I'm curious what modern animation systems use quaternions. Can anyone who knows graphics well provide references?
- mrcode007 3y ago
  They’re ubiquitous in computer graphics for video games and to some degree in space and aeronautics. They’re better than Euler angles because they don’t go singular as often. There are other more ansvanced and better parametrizations though. In space rotations are of critical importance and ensure stability or instability depending on the axis of rotation
  - interroboink 3y ago
    > they don’t go singular as often A term I've often heard is "gimbal lock"[1]. Can happen, eg, when a player points the camera straight up, then tries to rotate themselves (depending on how your axes are set up). I don't believe quaternions _ever_ have that particular problem, so not quite sure what is meant by "less often," here. [1] https://en.wikipedia.org/wiki/Gimbal_lock https://en.wikipedia.org/wiki/Gimbal_lock
    - mrcode007 3y ago
      You’re right, quaternions don’t go singular. I misremembered what the problem was. And the problem with the quaternions is that the multiple revolutions will resolve to the same quaternion and then you don’t know how many times you’ve revolved around. For some applications this is important to know.
  - j2kun 3y ago
    References though. If you had to prove it was ubiquitous, what would you point to?
    - djmips 3y ago
      You could look at the Unreal Engine source.
admsmz 3y ago
Geometric Algebra subsumes and extends quaternions: https://www.youtube.com/watch?v=60z_hpEAtD8 https://www.youtube.com/watch?v=60z_hpEAtD8
gatane 3y ago
The answer to slerp is to slerp over slerp
teleforce 3y ago
Matlab quaternion interpolation function based on this paper: https://www.mathworks.com/help/aerotbx/ug/quatinterp.html https://www.mathworks.com/help/aerotbx/ug/quatinterp.html "qi=quatinterp(p,q,f,method) calculates the quaternion interpolation between two normalized quaternions p and q by interval fraction f. p and q are the two extremes between which the function calculates the quaternion. Aerospace Toolbox uses quaternions that are defined using the scalar-first convention."

63 points by teleforce 3y ago

iTokio 3y ago

1998

tbm57 3y ago

These were considered to be satanic black magic when they were first discovered. this is the 'quaternomicon'

teleforce 3y ago
Oliver Heaviside and Lord Kelvin both considered it as an evil concept and strongly believed that Maxwell was misled by using it to derive Maxwell's Equations. Heaviside then come up with the famous modern version of Maxwell's Equations with four equations but now ironically all of them can be represented with one concise single equation using quaternion based on geometric algebra [1]. It's probably worth to mention that Einstein was using quaternion concept when discovering his infamous special relativity [2]. It's interesting to note that complex number was once had the same fate as quaternion and it was also considered as an evil concept at the time by many well-known mathematicians not unlike the concept of zero/sifar/cipher when it's first invented and then popularized by the Arab mathematicians. Modern engineering achievements for example the wonderful world of mobile and digital communication are all based on complex number concept namely in-phase and quadrature or I/Q for short. This venerable concept of I/Q however is limited once you wan to include and consider polarization of the EM wave and this is when quaternion become very handy and useful [3]. I'd not be surprised that the next generation of wireless mobile i.e. 6G will be based on this new 3D quaternion based I/Q or I/Q/P (not yet a concept maybe soon), similar to the impact of I/Q as the basis of successful 2G to the 5G with their OFDM. [1] A derivation of the quaternion Maxwell’s equations using geometric algebra: http://peeterjoot.com/2018/03/05/a-derivation-of-the-quaternion-maxwells-equations-using-geometric-algebra/ http://peeterjoot.com/2018/03/05/a-derivation-of-the-quatern... [2] Quaternions in University-Level Physics Considering Special Relativity: https://arxiv.org/ftp/physics/papers/0308/0308017.pdf https://arxiv.org/ftp/physics/papers/0308/0308017.pdf [3] Modeling dual polarization wireless fading channels using quaternions: https://ieeexplore.ieee.org/document/1708024 https://ieeexplore.ieee.org/document/1708024

sxp 3y ago

It's somewhat silly to read about animation using a static doc. My preferred intro to quaternions in the context of motion is the pair of articles at https://acko.net/tag/quaternions/ https://acko.net/tag/quaternions/. They were critical helping me learn quaternions for computer graphics. I also recommend https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ as a primer if you aren't comfortable with regular complex numbers.

Culonavirus 3y ago
That site is pretty amazing.

j2kun 3y ago

I'm curious what modern animation systems use quaternions. Can anyone who knows graphics well provide references?

mrcode007 3y ago
They’re ubiquitous in computer graphics for video games and to some degree in space and aeronautics. They’re better than Euler angles because they don’t go singular as often. There are other more ansvanced and better parametrizations though. In space rotations are of critical importance and ensure stability or instability depending on the axis of rotation
- interroboink 3y ago
  > they don’t go singular as often A term I've often heard is "gimbal lock"[1]. Can happen, eg, when a player points the camera straight up, then tries to rotate themselves (depending on how your axes are set up). I don't believe quaternions _ever_ have that particular problem, so not quite sure what is meant by "less often," here. [1] https://en.wikipedia.org/wiki/Gimbal_lock https://en.wikipedia.org/wiki/Gimbal_lock
  - mrcode007 3y ago
    You’re right, quaternions don’t go singular. I misremembered what the problem was. And the problem with the quaternions is that the multiple revolutions will resolve to the same quaternion and then you don’t know how many times you’ve revolved around. For some applications this is important to know.
- j2kun 3y ago
  References though. If you had to prove it was ubiquitous, what would you point to?
  - djmips 3y ago
    You could look at the Unreal Engine source.

admsmz 3y ago

Geometric Algebra subsumes and extends quaternions: https://www.youtube.com/watch?v=60z_hpEAtD8 https://www.youtube.com/watch?v=60z_hpEAtD8

gatane 3y ago

The answer to slerp is to slerp over slerp

teleforce 3y ago

Matlab quaternion interpolation function based on this paper: https://www.mathworks.com/help/aerotbx/ug/quatinterp.html https://www.mathworks.com/help/aerotbx/ug/quatinterp.html "qi=quatinterp(p,q,f,method) calculates the quaternion interpolation between two normalized quaternions p and q by interval fraction f. p and q are the two extremes between which the function calculates the quaternion. Aerospace Toolbox uses quaternions that are defined using the scalar-first convention."