Talk:Dual quaternion

message: 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
:ek = e(ij) = (ei)j = −(ie)j = −i(ej) = ije = ke.
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 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.Rgdboer (talk) 22:56, 13 August 2008 (UTC)
replies: text: 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.Rgdboer (talk) 23:40, 22 September 2008 (UTC)
thread_title:  Limits to anti-commutativity 

message: I would like to make some changes to this article to make it easier to read and perhaps more useful. Prof McCarthy (talk) 03:45, 30 June 2011 (UTC)
thread_title:  Revisions to this article 

message: 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 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. Prof McCarthy (talk) 14:40, 12 August 2011 (UTC)
thread_title:  Conjugate of a quaternion 

message: 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 norm to avoid confusion.  — Preceding unsigned comment added by 2A00:1398:200:200:2677:3FF:FE1F:6278 (talk) 11:10, 19 February 2014 (UTC)
replies: text: No, it is not zero, but a (non-invertible) dual number (there are three “conjugations” in this algebra, see the section above). The article does not claim this “norm” makes a normed vector space.  Dual quaternions are the quaternion algebra over the dual number ring, and this “norm” reflects such structure, as well as convenient properties of the square root operation on dual numbers. Incnis Mrsi (talk) 18:10, 19 February 2014 (UTC)
thread_title:  Norm or Seminorm 

message: The sentence
is not meaningful unless the article specifies whether its convention for the dual Quaternion's rigid transform is T*R or R*T.--2A02:8084:2562:5E80:9ED0:F6E5:50EF:B820 (talk) 11:39, 21 October 2018 (UTC)
thread_title:  Is rotation or displacement applied first? 

message: The 8-dimensional real representation is spelled out. Are there higher-dimensional representations (that are not simple)? Are there complex representations? Can the 8-D rep be written with Pauli matrices, and what would it look like? 67.198.37.16 (talk) 20:44, 21 May 2024 (UTC)
thread_title:  Higher-dimensional representations?