Talk:Applications of dual quaternions to 2D geometry

Unmotivated

This article purports to provide a representation of Euclidean plane isometry with the dual-complex algebra (which has no published sources, only arXiv references). In fact, complex numbers are sufficient to represent such mappings as is shown at Euclidean plane isometry#Isometries in the complex plane. A blog article by Terence Tao also is cited but Tao’s results are not, and it is just a blog article, not WP:Reliable source. Is this worthy encyclopedia material ? — Rgdboer (talk) 22:27, 9 September 2019 (UTC)

BTW I doubt this should exist as a separate article at all. It is a subalgebra of dual quaternions and does not differ from it qualitatively. Merge to? Incnis Mrsi (talk) 04:43, 10 September 2019 (UTC)

The dual-complex numbers express all rigid-body motions as rotations about some point. If the point is at infinity, then the rotations turn into translations. The complex number formalism that you gave instead represents rigid body motions as: rotations about a fixed origin, followed by a translation. This is a difference between the formalisms. Note that the way that the projective plane with its points at infinity occurs naturally in the Dual-complex numbers should be of interest. Additionally, dual-complex numbers may have applications in linearly interpolating between two rigid-body motions using an algorithm similar to SLERP. The fact that taking a logarithm of a dual-complex number is so simple implies that SLERP can be adapted to it quite straightforwardly. One of the papers that I cited appears to do this in the context of image processing. --Svennik (talk) 11:47, 10 September 2019 (UTC)

I've switched two of the Arxiv references for journal articles. For yet more evidence of notability, see the article on Eduard Study. Thanks. --Svennik (talk) 00:09, 14 September 2019 (UTC)

Title mismatch cross-post

See Wikipedia talk:WikiProject Mathematics#Article and talk page title mismatch for Dual-complex numbers. Comments should be made there, not here. GeoffreyT2000 (talk) 22:07, 4 October 2019 (UTC)

Missing the "odd" counterpart to the dual-complex numbers

The dual-complex numbers are the even part of the dual quaternions when the latter are seen as a Clifford algebra. The corresponding odd part also has applications in planar geometry. This odd counterpart can be used to represent:

• Reflections
• Glide reflections
• Line objects

Additionally, the geometric construction given in this article can be generalised to the odd counterpart.

I feel that this would serve as an excellent example of the even/odd duality in Clifford algebra.

It would be nice to expand this article to include this additional information. --Svennik (talk) 11:51, 20 December 2019 (UTC)

All online sources call "dual-complex numbers" a different thing

They call it the thing described here in the section "terminology". Should not this article be moved?--Reciprocist (talk) 04:37, 24 June 2022 (UTC)

Maybe. I'm not sure whether this "number system" has many uses presently except as a subalgebra of PGA or the dual quaternions. Its applications to planar geometry are more complete when generalised to these larger algebras. See PGA (Projective Geometric Algebra). --Svennik (talk) 10:36, 24 June 2022 (UTC)

Any ideas on how this thing should be properly called? Possibly, it should be a subsection of Dual quaternions?--Reciprocist (talk) 14:55, 26 June 2022 (UTC)

The article on dual quaternions is quite long already. We could have a separate article on either:

- applications of dual quaternions to 2D geometry

- 2D PGA (Projective Geometric Algebra)

These two Clifford algebras are non-isomorphic but very similar, so I'm not fully decided on which would be better. 2D PGA as an algebra is isomorphic to the algebra of $2\times 2$ dual number matrices, which is discussed extensively in the article on Laguerre transformations, though not using the standard notation and perspective of PGA.

I'm somewhat partial to expanding this article to "applications of dual quaternions to 2D geometry", and maybe writing it not in a Geometric Algebra style (meaning avoiding too much discussion of multivectors and exterior products). The idea is that once you "accept" that quaternions can represent rotational symmetries of a sphere, and that dual numbers can represent infinitesimals, then an infinitesimal neighbourhood of a sphere can be made to behave like the Euclidean plane. The dual quaternions then provide a means of rotating, translation and reflecting this plane. I could leave stubs for people to have discussions in GA terms (that is, featuring multivectors and the exterior product). I'm curious to hear other people's thoughts. Svennik (talk) 20:11, 27 June 2022 (UTC)

That would be great! i would support renaming this article to "applications of dual quaternions to 2D geometry", and it would be nice to have an article on true dual-complex numbers.--Reciprocist (talk) 08:43, 30 June 2022 (UTC)

I think maybe we should call them the planar quaternions for now. Your edits are making the article more confusing. I've chosen the name "planar quaternions" because I've seen it in a YouTube tutorial on PGA. Honestly, this algebra has many names, and "dual-complex numbers" is probably the most damaging one, given that there might be good reasons to use the algebra $\mathbb {C} [\varepsilon ]/(\varepsilon ^{2})$. Here's a video by a well-known educator and professor on the "dual-complex number" system in this sense (which is the one you're promoting): https://www.youtube.com/watch?v=dNpjzgkYWVY --Svennik (talk) 18:41, 1 July 2022 (UTC)

I have never hear the term "planar quaternions" but I am OK with it. I agree that "dual-complex numbers" is indeed damaging.--Reciprocist (talk) 00:29, 5 July 2022 (UTC)