Talk:Applications of dual quaternions to 2D geometry: Difference between revisions
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: BTW {{serif|I}} doubt this should exist as a separate article at all. It is a subalgebra of [[dual quaternion]]s and does not differ from it qualitatively. Merge to? [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 04:43, 10 September 2019 (UTC) |
: BTW {{serif|I}} doubt this should exist as a separate article at all. It is a subalgebra of [[dual quaternion]]s and does not differ from it qualitatively. Merge to? [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 04:43, 10 September 2019 (UTC) |
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:: @[[User:Incnis Mrsi|Incnis Mrsi]]: The article on [[dual quaternions]] is complicated enough as it is. I am against this. --[[User:Svennik|Svennik]] ([[User talk:Svennik|talk]]) 10:38, 10 September 2019 (UTC) |
:: @[[User:Incnis Mrsi|Incnis Mrsi]]: The article on [[dual quaternions]] is complicated enough as it is. I am against this. --[[User:Svennik|Svennik]] ([[User talk:Svennik|talk]]) 10:38, 10 September 2019 (UTC) |
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: 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 translation. 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. --[[User:Svennik|Svennik]] ([[User talk:Svennik|talk]]) 11:47, 10 September 2019 (UTC) |
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:: LoL. Mathematicians merely use other terminology, such as [[affine transformation]]s—over [[complex number|ℂ]] for this case—instead of mysteriously sounding “rotations about points at infinity”. But [[user:Svennik|the cool computer scientist]] may deem mathematicians ignorant. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 10:55, 10 September 2019 (UTC) |
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::: What you're not getting, [[User:Incnis Mrsi|Incnis Mrsi]], is that translations are a limiting case of rotations. Dual-complex numbers make this explicit. And don't put words in my mouth: ''may deem mathematicians ignorant''. Don't ''ever'' attribute things to me that I never said. --[[User:Svennik|Svennik]] ([[User talk:Svennik|talk]]) 10:59, 10 September 2019 (UTC) |
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:::: Risible. {{serif|I}} couldn’t obtain my diploma being unable to demonstrate that a sequence of rotations can be made to converge to a translation by sending centres (fixed points) of rotations to infinity. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 11:18, 10 September 2019 (UTC) |
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::::: I think this discussion has turned heated. I'm fully to blame. Apologies to you and Rgdboer. If you're willing, I may edit this discussion down to my claim that [[Dual-complex numbers]] express rigid-body motions from a different perspective (''not'' rotations then translations), while leaving yours and Rgdboer's original comments. --[[User:Svennik|Svennik]] ([[User talk:Svennik|talk]]) 11:26, 10 September 2019 (UTC) |
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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)
- @Incnis Mrsi: The article on dual quaternions is complicated enough as it is. I am against this. --Svennik (talk) 10:38, 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 translation. 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. --Svennik (talk) 11:47, 10 September 2019 (UTC)