Visualizing quaternions: An explorable video series (2018)

by uncircle • 12 months ago

85 13 12 months ago

dang• 11 months ago

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• 11 months ago

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

calebm• 11 months ago

Wow, this is very cool.

JKCalhoun• 11 months ago

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.

srean5• 11 months ago

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.

Koshkin• 11 months ago

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

srean5• 11 months ago

Thanks a bunch

xeonmc• 11 months ago

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.

srean5• 11 months ago

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

jacobolus2• 11 months ago

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...

srean5• 11 months ago

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.

Visualizing quaternions: An explorable video series (2018)

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