4 ms·
Visualizing quaternions: An explorable video series (2018)
- dang 1y agoRelated. Others? 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)
- gnabgib 1y agoShow HN: Visualizing the math that powers 3D character animation (238 points, 2022, 53 comments) https://news.ycombinator.com/item?id=31724613 https://news.ycombinator.com/item?id=31724613
- calebm 1y agoWow, this is very cool.
- JKCalhoun 1y agoVery 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.
- srean 1y agoQuestions 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 1y agoI am not familiar with this area, but see, e.g. https://link.springer.com/book/10.1007/978-3-0348-0622-0 https://link.springer.com/book/10.1007/978-3-0348-0622-0 There are also these notes: https://www.geometrictools.com/Documentation/Quaternions.pdf https://www.geometrictools.com/Documentation/Quaternions.pdf
- srean 1y agoThanks a bunch
- xeonmc 1y agoA 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 1y agoHeard 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
- jacobolus 1y agoConformal 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_mappings) https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...
- srean 1y agoYes 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 1y agoYou're probably looking for something like Sudbery 1977, https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-analysis-memo.pdf https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana... (published 1979, doi: 10.1017/S0305004100055638)
- srean 1y agoFantastic. Thanks for the reference.