Re: [math-fun] Gimbal lock??

[Fred Lunnon]

Following thread started by Henry Barber on the math-fun list ---

On 1/29/20, Fred Lunnon <fred....@gmail.com> wrote:
> Good luck with convincing the engineers involved!
> Mind you, it is only getting on for 180 years ago that Hamilton
> came up with quaternions. So early days ... er, centuries?
>
> But perhaps it might be more constructive to attempt to
> understand why there is such instinctive resistance among
> engineers to the whole notion of quaternions, never mind
> more general geometric (Clifford) algebra.
>
> There's a frustrating pedagogical phenomenon involved in
> such investigations. While I can well recall my own sense of
> bewilderment on first encountering Hestenes' early book on
> these matters, I cannot muster the slightest insight into the
> cause of those conceptual difficulties. As a result, I now am
> stranded as far away from offering assistance to the uninitiated
> as I earlier was from receiving any: I simply cannot understand
> _why_ they can't understand.
>
> Anyway, here's a couple of possible clues: perhaps others
> can come up with more suggestions.
>
> (A) It's noteworthy that the first thing Heaviside did was to
> dissect Hamiton's elegant unity into "scalar" & "vector" parts,
> which went on to gain pretty much universal acceptance.
> There seems to be a mental hurdle in human minds obstructing
> the union of disparate familiar categories under a common
> umbrella: in this case, familiarity with angles & Cartesian
> coordinates actively obstructs conceptualisation of quaternions.
>
> (B) It is rarely made explicit that (like vectors) quaternions
> come in two flavours: polar & axial, depending on application.
> It seems that Hamilton himself was confused over this,
> which contributed to early controversy about their validity.
> There's a informative but slightly muddled paper on this topic
> somewhere on the internet which proceeds from the quaint
> premiss that they must exclusively be one or the other, despite
> the arguments put forward clearly illustrating a dichotomy.
>
> WFL
>
>
>
> On 1/28/20, Henry Baker <hba...@pipeline.com> wrote:
>> Perhaps it's time to learn quaternions??
>>
>> From comp.risks:
>>
>> Date: Fri, 10 Jan 2020 20:24:07 +0000
>> From: "Clive D.W. Feather" <cl...@davros.org>
>> Subject: Boeing 737s can't land facing west (FAA)
>>
>> "The FAA received reports earlier this year of three incidents of display
>> electronic unit (DEU) software errors on Model 737 NG airplanes flying
>> into
>> runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a
>> selected instrument approach to a runway with a 270-degree true heading,
>> and
>> all six DUs stayed blank until a different runway was selected. [...]
>> The
>> investigation revealed that the problem occurs when this combination of
>> software is installed and a susceptible runway with a 270-degree true
>> heading is selected for instrument approach. Not all runways with a
>> 270-degree true heading are susceptible; only seven runways worldwide, as
>> identified in this AD, have latitude and longitude values that cause the
>> blanking behavior."
>>
>> (Note that this is all 6 displays on each plane, not 2 displays on each
>> of
>> three planes.)
>>
>> The runways in question are:
>>
>> Runway 26, Pine Bluffs, Wyoming, USA (82V)
>> Runway 28, Wayne County, Ohio, USA (KBJJ)
>> Runway 28, Chippewa County, Michigan, USA (KCIU)
>> Runway 26, Cavern City, New Mexico, USA (KCNM)
>> Runway 25, Barrow, Alaska, USA (PABR)
>> Runway 28, La Mina, La Guajira, Colombia (SKLM)
>> Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
>>
>> (The numbers are magnetic bearings, whereas the problem is apparently
>> related to true bearing.)
>>
>> Original FAA notice:
>> <http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc27b862584dd005c1a60/$FILE/2019-25-17.pdf>
>>
>>
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>>
>

[garret sobczyk]

Clifford's geometric algebra should be recognized as a general (multi)vector analysis of higher dimensional vector spaces. It should also be recognized as an integral part of real and complex (square) matrix algebras of dimensions 2^n x 2^n. The simplest case is when n=1, the case of 2x2 real and complex matrices. The next simplest case is when n=2, the case of 4x4 real and complex matrices. Once the reader understands these two important special cases, which cover the important geometric algebras of 3 dimensional Euclidean space and the 4 dimensional Dirac algebra of spacetime, referred to by Hestenes as spacetime algebra, all other geometric algebras can be constructed by taking Kronecker products of successive copies of the 2x2 real and complex matrix algebras. This approach to geometric algebra is systematically explored in my new book, "Matrix Gateway to Geometric Algebra, Spacetime and Spinors",

https://www.amazon.com/s?k=Garret+Sobczyk&ref=nb_sb_noss_2

A short introduction to "Geometric Matrices" of 2x2 and 4x4, is provided in the appendix of my newest article, "Notes on Plucker's relations in geometric algebra", to appear in Advances in Mathematics 363 (2020) 106959. Elsevier has kindly provided a temporary free access link to this article, which can be found at the link: https://www.garretstar.com/secciones/publications/publications.html

    

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