Visualizing quaternions: An explorable video series (2018)

[uncircle]

https://eater.net/quaternions

[dang]

Related. Others?

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)

[gnabgib → dang]

Show HN: Visualizing the math that powers 3D character animation (238 points, 2022, 53 comments) https://news.ycombinator.com/item?id=31724613

[srean]

Questions for mathematicians out here.

Is there such a thing as quaternion analysis -- calculus of functions from quaternions to quaternions.

What would be their key theorems ? What would be the analogue of conformal mappings, if any ?

Any book recommendations would be gratefully appreciated.

[jacobolus → srean]

Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations. The 2 dimensional case is special.

See: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...

[srean → jacobolus]

Yes of course, but I am curious about any interesting structures that functions from quaternion to quaternion may possess. I used conformal mapping as an example of an interesting structure. I could have used Cauchy Riemann as another example.

[jacobolus → srean]

You're probably looking for something like Sudbery 1977,

https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana...

(published 1979, doi: 10.1017/S0305004100055638)

[srean → jacobolus]

Fantastic. Thanks for the reference.

[Koshkin → srean]

I am not familiar with this area, but see, e.g. https://link.springer.com/book/10.1007/978-3-0348-0622-0

There are also these notes: https://www.geometrictools.com/Documentation/Quaternions.pdf

[srean → Koshkin]

Thanks a bunch

[xeonmc → srean]

A quaternion encodes uniform scaling + rotation. The logarithm of a quaternion is its axis-angle-nepers form, and vice versa.

    quat = sqrt( exp( nepers + radians * <axis> ) )

So I think with this exponential map, the rest of its calculus can be extended from that.

[srean → xeonmc]

Heard the word 'nepers' after many decades. Are you by any chance an Electrical major ?

Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.

Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions

[JKCalhoun]

Very cool.

On MacOS: no audio on Safari, worked in Chrome though.

I was odd to me to see that Ben Eater was involved. His talents are broad.

[calebm]

Wow, this is very cool.