[HN Gopher] Quaternion Differentiation (2012)
___________________________________________________________________
Quaternion Differentiation (2012)
Author : niborgen
Score : 102 points
Date : 2024-06-28 12:22 UTC (1 days ago)
(HTM) web link (fgiesen.wordpress.com)
(TXT) w3m dump (fgiesen.wordpress.com)
| ykonstant wrote:
| The general theory of this differentiation process is given here:
| https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)
| bbor wrote:
| A) This is one of the best pieces of math pedagogy I've seen in
| ages. A true "explanation" instead of a "reference" -- thanks for
| sharing.
|
| B) I think we should pass a hacker news law via referendum
| (that's a thing, right? It should be!) that any article
| mentioning quaternions must also generalize to Octonions, or at
| least gesture in that vague direction to pay respect. As the
| first paper I found on Octonion Differentiation says best:
| Each Cayley-Dickson algebra Ar+1 is obtained from the preceding
| Ar with the help of the so called doubling procedure [1, 14, 17].
| This gives the family of embedded algebras: Ar -- Ar+1 -- ....
| For a unification of notation it is convenient to put: A0 = R for
| the real field, A1 = C for the complex field, A2 = H denotes the
| quaternion skew field, A3 = O is the octonion algebra, A4 denotes
| the sedenion algebra. The quaternion skew field is
| associative, but non-commutative. The octonion algebra is the
| alternative division algebra with the multiplicative norm. The
| sedenion algebra and Cayley-Dickson algebras of higher order r >=
| 4 are not division algebras and have not any non-trivial
| multiplicative norm. Each equation of the form ax = b with non-
| zero octonion a and any octonion b can be resolved in the
| octonion algebra: x = a-1b, but it may be non-resolvable in
| Cayley-Dickson algebras of higher order r >= 4 because of
| divisors of zero. Therefore, in this article
| differential equations are considered [only] with octonion or
| quaternion variables for octonion or quaternion valued functions.
|
| https://arxiv.org/pdf/1003.2620 FWIW the octonion answer seems to
| be much more dependent on what kind of analysis you're doing. AKA
| "it's complicated"
| archgoon wrote:
| (B) Would exclude this article so I think it's a bad rule.
| JadeNB wrote:
| > I think we should pass a hacker news law via referendum
| (that's a thing, right? It should be!) that any article
| mentioning quaternions must also generalize to Octonions ....
|
| Given how much weirder octonions are than quaternions, I think
| that's mainly a recipe for not learning the most interesting
| things about quaternions. I'd rather hear about the connections
| between octonions and exceptional groups on one hand, and let
| quaternions be good at what they're good at on the other hand,
| and let the twain meet only as circumstances dictate.
| aap_ wrote:
| Why octonions though? Quaternions are nice and simple,
| octonions are much stranger. Still hoping we find interesting
| uses for them sometime.
| superposeur wrote:
| The "differentiation" in the title turns out to be derivative of
| a quarternion-valued function with respect to a scalar parameter.
|
| But, I wonder if you math folks know of a definition of
| derivative of a quarternionic function with respect to
| quarternionic variable, generalizing the Cauchy-Riemann
| definition [1] of complex differentiation?
|
| [1]
| https://en.m.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equat...
| ziofill wrote:
| In Wirtinger calculus
| (https://en.m.wikipedia.org/wiki/Wirtinger_derivatives) you
| consider a complex variable and its conjugate as independent.
| This simplifies a lot of things e.g. Cauchy Riemann becomes
| just df/dz* = 0.
|
| TensorFlow works this way, jax instead differentiates real and
| imaginary parts.
|
| I wonder if there is a version for quaternions now.
| hbrav wrote:
| The problem with generalizing this to quarternions is that
| the conjugation operation for quaternions can be expressed
| using arithmetic operations on the quaternion:
|
| q* = -0.5 _(q + iqi + jqj + kqk)
|
| So the analogy to complex analysis where we'd talk of z and
| z_ as independent doesn't work anymore - since we can write
| q* as an 'analytic' function of q.
|
| It's not surprising you'd need something different though,
| since (q, q*) is only two variables and quaternions are
| 4-dimensional. I don't know a lot about quaternions, but
| Penrose introduces them in The Road to Reality and says
| (roughly) "yeah, they don't have the nice analytic-function
| properties that complex numbers have" and seems to kinda
| leave it at that. If anyone knows more and wants to reduce my
| ignorance, I'd be grateful.
| BenoitP wrote:
| In the "Multivector Derivative" section you may plug in an
| arbitrary multivector function of your liking:
|
| https://en.wikipedia.org/wiki/Geometric_calculus
| edflsafoiewq wrote:
| The Cauchy-Riemann equations just say that the derivative (as a
| map of 2-vectors) acts as a complex multiplication (treating
| 2-vectors as complex numbers). For quaternions you would say it
| acts as quaternion multiplication I guess. But since
| quaternions aren't commutative, you would also have to say if
| it's acting as a left or a right multiplication...
| bananaflag wrote:
| https://en.m.wikipedia.org/wiki/Quaternionic_analysis
___________________________________________________________________
(page generated 2024-06-29 23:01 UTC)