Related: https://eater.net/quaternions
What a beautiful and high quality presentations. All the praises to the author.
Quaternions and spherical trigonometry
11–20 of 53 posts
Re: Quaternions and spherical trigonometry
#12Related: https://eater.net/quaternions
wow love the fact that you can interact with the videos
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.
Re: Quaternions and spherical trigonometry
#13Related: https://eater.net/quaternions
Related:
Visualizing quaternions (2018) - 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 - April 2022 (15 comments)
Visualizing quaternions: An explorable video series - https://news.ycombinator.com/item?id=18310788 - Oct 2018 (32 comments)
Re: Quaternions and spherical trigonometry
#14[flagged]
[deleted]
Re: Quaternions and spherical trigonometry
#15Related: https://eater.net/quaternions
I still gotta grasp this for IMUs
Re: Quaternions and spherical trigonometry
#16Just 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?
Re: Quaternions and spherical trigonometry
#17Just 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…
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-ar...
Re: Quaternions and spherical trigonometry
#18Just 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 really like the welch labs series on imaginary numbers which covers the first part of what you talk about -- leveling up the notion of what a complex number is. Though his focus was more on solving simple equations with no real roots, but really detailing how/what is really going on.
It is a great precursor to then thinking about quaternions
https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
Re: Quaternions and spherical trigonometry
#19Just 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…
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-ar...
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.
Re: Quaternions and spherical trigonometry
#20Just 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'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?
Octonions and up (more generally known as hypercomplex numbers) exist, but every time you pull the "double dimensions by adding more imaginary components" trick[0], you lose another useful property.
Real to complex loses total ordering. Complex to quaternion loses commutativity. Quaternion to octonion loses associativity (but they are at least alternative). The sedenions aren't even alternative, and they have zero divisors to boot.
You can also generalize hypercomplex numbers to the study of Clifford algebras.
[0] The Cayley-Dickson construction