Quaternions and Video Game Design

[Ben2]

On p. 566 (Section 563) of Hamilton's Lectures on Quaternions, I find "Operating with phi, and making reductions analogous to those of recent articles..." The 2nd derivation, beginning with "phi rho prime" so far totally eludes me.
Let me apologize if this is the wrong forum. Quaternions have apparently been applied in both the unified-force problem and videogame design. I've tried unsuccessfully to get help from Hamilton's home university (Trinity College in Ireland), and Hamilton's auditors are of course long gone.
Any help or references will be greatly appreciated. Thanks.

 

[jedishrfu]

I don't know if this will help but 3blue1brown has created some videos on Quaternions:



There are other references on the web that may answer your question:

http://web.cs.iastate.edu/~cs577/handouts/quaternion.pdf

https://www.quora.com/What-is-the-first-and-second-derivative-of-quaternion

 

[Mark44]

On p. 566 (Section 563) of Hamilton's Lectures on Quaternions, I find "Operating with phi, and making reductions analogous to those of recent articles..." The 2nd derivation, beginning with "phi rho prime" so far totally eludes me.

I'd venture to say that most of us here don't have this book, so aren't familiar with what you're asking.

Can you upload a clear photo of this page? That would be helpful.

 

[fresh_42]

Here's a link: (attention, it's 42 Mb)

https://ia801407.us.archive.org/4/items/lecturesonquater00hami/lecturesonquater00hami.pdf

 

[jedishrfu]

Awesome, @fresh_42, Hamilton’s voice reverberates once again through the ages, saying I knew you’d find my mathematical friends useful.

 

[Ben2]

<Moderator's note: Merged as it is still the same topic and even reference.>

My specific request is for help with proofs of equations for \phi\rho' or \phi\rho'' in Section 563 (p. 566) of
Hamilton's Lectures on Quaternions. I can provide certain details should the readership desire them.
I've received no hints or advice from any other source, to include Hamilton's home university, Trinity College in Dublin, Ireland. Quats have apparently been used in videogame design, and one researcher into the unified-force problem has told me he's been unable to study certain aspects of that question using other
techniques (e.g. differential geometry).
Should this query be considered inappropriate for physicsforums.com, please accept my apologies. Otherwise will be grateful for any advice received. As always, my special thanks to the moderators.
Ben

 

[jedishrfu]

We can't help you unless we see what is on pg 566 of Section 563 of Hamilton's Lectures on QUaternions.

IS this what you're referring to?

https://archive.org/details/lecturesonquater00hami/page/566

 

[Ben2]

Jedishrfu,
That is precisely right. The fourth and second equations from foot of p. 566 are the subjects of my question.
The difficulty is "making reductions analogous to those of recent articles..." Again, I'll be grateful for any help
available, and thanks especially for your prompt reply!
Ben

 

[steenis]

Here are some references of books about quaternions:

Hanson - Visualizing Quaternions (2006)
Kuipers - Quaternions and Rotation Sequences (1999)
Vince - Quaternions for Computer Graphics (2011)
Vince - Rotation Transforms for Computer Graphics (2011)​

If you are into computers graphics (games and so) you might find all the books of John A. Vince interesting:

Vince - Calculus for Computer Graphics (2013)
Vince - Geometric Algebra for Computer Graphics (2008)
Vince - Geometry for Computer Graphics; formulae, examples and proofs (2005)
Vince - Mathematics for Computer Graphics (5th edition,2014)
Vince - Vector Analysis for Computer Graphics (2007)​

and many more of Vince …

 

[jedishrfu]

I'm afraid I can't help much here. Hamilton's notation seems very dated from what we use today (meaning I can't figure it out). His use of the words "articles" I think refers to simply prior numbered section of his book and whatever transformations he's made earlier.

I looked for something that had the words "inverse quaternions general linear and vector equation" and came up with these paper of theorems and proofs of quaternions and matrices of quaternions:

https://core.ac.uk/download/pdf/82180866.pdf

https://www.researchgate.net/publication/225894480_Linear_Quaternionic_Equations_and_Their_Systems

https://www.sciencedirect.com/science/article/pii/S1018364717303130​

and lastly, Wikipedia for the references the article provides:

https://en.wikipedia.org/wiki/Quaternion​

Quaternions in Computer Science

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

https://pdfs.semanticscholar.org/f66c/ac96440da71212871f662f916ede4aee825a.pdf

https://studylib.net/doc/18121734/quaternion-and-rotation---department-of-computer-science​

 

[Ben2]

Jedishrfu:
Again you're correct; Hamilton's notation is brutal, but provides an alternative to "coordinatization" as in Shpakivskyi's work (2nd reference).
Thanks for tracking down these tremendous references!
Ben