Quaternion Differentiation (2012) • Hacker News Board

Re: Quaternion Differentiation (2012)

#11

Post by edflsafoiewq » Sat, Jun 29, 2024, 9:46 AM UTC

superposeur wrote: post #6

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...

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...

Re: Quaternion Differentiation (2012)

#12

Post by bananaflag » Sat, Jun 29, 2024, 10:41 AM UTC

superposeur wrote: post #6

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...

https://en.m.wikipedia.org/wiki/Quaternionic_analysis

Re: Quaternion Differentiation (2012)

#13

Post by hbrav » Sat, Jun 29, 2024, 3:55 PM UTC

ziofill wrote: post #8

superposeur wrote: post #6

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...

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.

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.

Re: Quaternion Differentiation (2012)

#14

Post by alasarmas » Sun, Jun 30, 2024, 12:47 AM UTC

hbrav wrote: post #13

ziofill wrote: post #8

Earlier quoted context omitted.

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.

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,…

I think some of the stars for your complex conjugates fell foul of HN formatting:

https://news.ycombinator.com/formatdoc