Talk:Quaternions and spatial rotation: Difference between revisions

Revision as of 16:28, 24 May 2024

Revision as of 17:22, 24 May 2024

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:I think the example of non-commutative rotations is good to have in the article, but I don't think the current explanation is clear enough. For someone who is good at thinking in spatial terms, it's easy to follow, but many people will not be, especially people who are new to quaternions. A diagram would really help here, maybe an animated GIF? Does anyone have the ability to make a good one? - [[User:Rainwarrior|Rainwarrior]] 20:09, 10 February 2007 (UTC)

I disagree with a lot of the intent in the above posts. Representing rotations with quaternions is a specific technical idea. I'm not saying that the article should be filled with jargon; merely that handwaving descriptions of rotating books won't be useful to someone who arrives at Wikipedia wanting to know how to rotate with quaternionsquarternions. That being said, the book analogy is good, but perhaps it should go at "Non-commutativity" itself, which currently redirects to commutativity. Add a little ''For a technical discussion of non-commutativity, see [[Commutativity]].'' line, and you have the perfect place to explain why Rubics Cube's are so difficult. [[User:Endomorphic|Endomorphic]] 20:27, 11 February 2007 (UTC)

== Application to 3D animation ==

This article is too technical for me. How are QuaternionsQuarternions used in 3D graphics? --[[User:24.249.108.133|24.249.108.133]] 22:59, 11 April 2007 (UTC)

== Great reference ==

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* Let f be an ''isometry'' (= linear map preserving the distance) of R3. ''Assume 1 to be an eigenvalue of f.'' Say, there exists a vector V such that f(V)=V. Then the line generated by V is stable under f. As f is an isometry, it preserves the orthogonality. Thus, f preserves the plane orthogonal to V. In particular, f restricts to an isometry of V. You probably know that isometries in dimension 2 are rotations or orthogonal symetries with respect to a line generated by W. Thus ''f is a rotation of axis RV (first case) or a symetry with respect to the generated by V and W.''

* {{u|MathsPoetry}} has partially right. There exists a beautiful geometric argument (beautiful for anyone who loves geometry). The field of quaternions '''H''' is equipped with an Euclidean metric |.|, and we have <math>|ab|=|a|\times |b|</math>. In particular, the action A(q) of a ''unitary quaternionquarternion'' q by conjugation preserves this Euclidean metric. Then, A(q) is an isometry of the Euclidean vector space '''H''', which preserves 1, then the line of reals and its orthogonal, which is the vector space ''E'' of the pure quaternions. Here ''E'' is a vector space of dimension 3, and can be naturally identified with R3 because '''i''', '''j''' and '''k''' is an orthonormal basis. The linear map A(q) restricts to an isometry of ''E'', still denoted by A(q). Now, the set of unitary quaternions is simply the unit sphere of '''H'''. So you can pass continuously from 1 to q. There is a continuous path of isometries from the identity to A(q). Hence, the determinant of A(q) is 1. In other words, A(q) is a rotation. ''This prrof is well-known, and it is not hard to find references, if necessary. I hope no.''

* The explicit computation ("your proof") is also very nice, but appears to me as a sort of trick. But it is useful to understand really how a unitary quaternion represents a rotation, with its axis and its angle.

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== Problem with example ==

In the example halfway down the page, i,j,k are used for the unit basis vectors for 3-d cartesian space, and also for the quaternionquarternion imaginary terms. These are not the same thing and it makes the example very confusing.[[User:Eregli bob|Eregli bob]] ([[User talk:Eregli bob|talk]]) 05:27, 22 September 2012 (UTC)

:I must agree with this comment. The two concepts, being the vector part of a quaternion and a vector in Euclidian space cannot be conflated into the same concept without comment. This can be seen from the fact that a quaternion unit vector squares to −1 (under the quaternion product), whereas a geometric unit vector squares to +1 (under the dot product). Both these are easily accommodated in a [[geometric algebra]] but this ephasizes that they are distinct; they are in fact each other's [[Hodge dual]] in a 3-D geometric algebra. Thus special care (and explanation) is needed for interpreting ''i'', ''j'' and ''k'' as geometric unit vectors. I do not know the traditional way of resolving/presenting this; I suspect it is by simply conflating the two, but reserving the notation ''q''² to mean the quaternion product rather than the vector dot product, for example for the pure vector ''u'', ''u''⋅''u'' = ''u''^∗''u'' = (−''u'')''u'' = −''u''². Whatever the case, it should be made clearer. — [[User_talk:Quondum|''Quondum'']] 11:04, 22 September 2012 (UTC)

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which is the same as rotating (conjugating) by&nbsp;{{math|'''q'''}} and then by&nbsp;{{math|'''p'''}}. The scalar component of the result is necessarily zero."

Shouldn't the operation in the above equation by conjugation (*), not inverse (-1)? One then needs to know that (pq)^* = q^*p^*. Also why is the scalar component of the result is necessarily zero? Since quaternionsquarternions are a division ring, any non-zero rotation can be represented as the product of two other rotations, so this would mean the scaler part of any rotation is zero, which can't be right.--[[User:ArnoldReinhold|agr]] ([[User talk:ArnoldReinhold|talk]]) 17:07, 4 August 2013 (UTC)</s>

:I understand now. For units, the inverse is the conjugate (I added a sentence to point this out), and the conjugation operation is being applied to a 3-vector, so scaler part remains zero.--[[User:ArnoldReinhold|agr]] ([[User talk:ArnoldReinhold|talk]]) 18:35, 8 August 2013 (UTC)