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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 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)reply
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.
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)reply
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 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)reply
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)reply
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 . 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)reply
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)reply
This article was reviewed by member(s) of WikiProject Articles for creation. The project works to allow users to contribute quality articles and media files to the encyclopedia and track their progress as they are developed. To participate, please visit the
project page for more information.Articles for creationWikipedia:WikiProject Articles for creationTemplate:WikiProject Articles for creationAfC articles
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of
mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join
the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics articles
This article is within the scope of WikiProject Numbers, a collaborative effort to improve the coverage of
Numbers on Wikipedia. If you would like to participate, please visit the project page, where you can join
the discussion and see a list of open tasks.NumbersWikipedia:WikiProject NumbersTemplate:WikiProject NumbersNumbers articles
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)reply
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.
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)reply
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 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)reply
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)reply
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 . 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)reply
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)reply