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Algebra theory that the construction of dual quaternions yields a dual unit multivector that commutes with the quaternion unit multi vectors---they are constructed as the even
Clifford algebra on four dimensional space with a degenerate (3,0) metric. The reason that I bring this up is that I would like to remove the specialized conjugates that have been introduced and focus on the usual and useful dual quaternion conjugate.
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In the article, a norm for dual-quaternions is defined. However, I was wondering whether this is a norm or a seminorm, because dual quaternions with a zero non-dual part would have the norm zero no matter what their dual part is, right? If I'm correct it might be better to write seminorm instead of
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dual part is also negated), in papers on kinematics. This initially caused me hours of confusion. I do not know if this is considered "correct", or if there is a better term. This seems to be so widespread that I think the reader should be informed/warned about this usage, with an explanation of
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I hope the issue of anti-commutation of the dual unit ε is resolved. There may be eight dimensional algebras constructed from ε, i, j, and k, in which anti-commutivity exists and associativity is lost, however, it is not correct to call them dual quaternions. It is possible to show using
Clifford
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The section refering to the skew theory algebra is marred by this problem. A separate article about a (revamped) appropriate algebra for the skew transformation is in order. Should associativity be dropped? Maybe only two non-commutativities are necessary. I realize that the dual quaternions are
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I commented out the section on dual quaternions for rigid motions because is claims that in this case the dual unit ε anti-commutes with the quaternion units i, j, k. This is not correct. It propagates a confusion introduced by Baker. The correct description is found in
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common fare for the mechanical engineers in some places. Nevertheless all mathematics must be consistent and there is a reason that the dual quaternion literature has not made it into standard mathematical texts. Expectations are too high.
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Also, the "conjugate" section seems to define two kinds of conjugates (in addition to the one I mention above), without naming them. Do these have names, such as "quaternion conjugate" and "dual conjuguate"?
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Checking the Martin Baker reference shows there is no claim of associativity; non-associativity is acknowleged with an example. For the time being this alternative dual quaternion idea is under consideration.
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I changed the notation to eliminate some of the unnecessary math formatting so it reads easier, and otherwise keep what was already there. However, I found some errors that needed to be corrected.
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why people use it, and a reference. If there is a better, standard, term for this quantity, I think it should be mentioned, since it seems to be widely used.
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It seems, when multiplication associates, that ε = e can only anti-commute with two of the i,j,k. For example, assume ei + ie = 0 and ej + je = 0. Then
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840:{\displaystyle {\vec {v}}'={\hat {q}}\cdot {\hat {v}}\cdot {\hat {q}}^{-1}={\hat {q}}\cdot 1\cdot {\hat {q}}^{-1}={\hat {q}}\cdot {\hat {q}}^{-1}=1}
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The section about "Dual quaternions and 4×4 homogeneous transforms" indicates this as the way to apply a dual transformation to a vector in
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I added the section on dual quaternions and spatial displacements, including formulas for the composition of spatial displacements.
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is not meaningful unless the article specifies whether its convention for the dual
Quaternion's rigid transform is T*R or R*T.--
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The 8-dimensional real representation is spelled out. Are there higher-dimensional representations (that are not
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over the dual number ring, and this “norm” reflects such structure, as well as convenient properties of the
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I would like to make some changes to this article to make it easier to read and perhaps more useful.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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I am not sure it is proper, but I increased the quality and importance ratings a little.
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This is not correct. It should be using the other conjugate, the one defined as
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109:and see a list of open tasks.
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901:. Dual quaternions are the
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