Talk:Axis–angle representation

Relation with rotation around a fixed axis

I don't see how the subject of this article differs from rotation around a fixed axis. For instance, is there anything which should be in one article and not the other? -- Jitse Niesen (talk) 04:23, 30 January 2007 (UTC)

While the axis angle representation of rotations does describe a rotation around a fixed axis, it is more commonly used to represent general rotations. More importantly, while that article discusses the physics involved in simple rotations, my intention with this article was to focus on how the axis angle representation of rotations is used in robot kinematics, its relationship to other representations of rotations, and its part in the more general concept of displacements. It might turn out that merging them is a better idea, but I'd like to try this as a seperate article first. Kborer 15:57, 30 January 2007 (UTC)

If you look at the axis angle section in rotation matrix it seems that a summary of what I was going to talk about in this article is there. I'd still like to have a more detailed explanation in this article. Kborer 18:42, 30 January 2007 (UTC)

Fair enough. Have a go at it and we'll see how it pans out. Sorry for my rash actions, but I hate having many short and abandoned articles on very similar topics. -- Jitse Niesen (talk) 02:14, 31 January 2007 (UTC)

Rename article

I propose that the name of this article be changed to "Rotation vector". Kborer (talk) 19:43, 28 December 2007 (UTC)

This name (axis/angle) is widely used in Robotics and Computer Vision for the unit-vector + angle representation. It seems that "rotation vector" is mainly used for the version where the vector magnitude expresses the rotation. Robertmacl (talk) 00:05, 24 September 2012 (UTC)

Notation for the antisymmetric matrix

I think the notation here is confusing. The 'hat' is so often used to mean a unit vector. Moreover, the usual notation for the relevant matrix is [w]_x. That is, the vector goes in square brackets and a subscript cross is added.

This notation is used, for example, in the definition of the relevant antisymmetric matrix in the article called 'Cross product'. I think for clarity and consistency that same notation ought to be used here.

Is there any special reason to use the 'hat' notation here that I am missing? —Preceding unsigned comment added by 55604PP (talk • contribs) 04:39, 27 January 2010 (UTC)

Whatever notation is used, the formula for exp(omega-tilda) we should remind the reader of the definition of omega as theta times u, e.g., by changing "exp(omega-tilda)" to "exp(omega-tilda) = exp(theta u-tilda). In the same formula, u-dot-L should be replaced by u-tilda (if the current tilda-notation is retained). — Preceding unsigned comment added by Tthrall (talk • contribs) 16:25, 27 June 2015 (UTC)

Simultaneous orthogonal rotation angle

The Simultaneous Orthogonal Rotation Angle seems to be exactly the same as the rotation vector (the preceding section), as near as I can tell. If you read the papers it seems that the authors also say that it is "a rotation vector", though in such an understated way that it doesn't make clear what their contribution is. If so, it isn't a new representation at all, although it is a new (to me) interpretation of the rotation vector, and reading it did make me think more about the usefulness of the rotation vector as a user-sensible representation. SORA is not exactly original research, since it has been published, but seems highly associated with two authors and has not been widely cited.

Robertmacl (talk) 23:47, 23 September 2012 (UTC)

I thought the same on reading this entry and the associated paper. The authors define the Simultaneous Orthogonal Rotation Angle as the product of angular velocity vector and a duration, which is clearly a standard rotation vector (in the same way that the product of angular acceleration and a duration is angular velocity). I am going to go ahead and remove this section.

64.106.20.136 (talk) 09:10, 19 February 2014 (UTC)

Yes, this had it's own article at one point but that was redirected here as a duplicate. See Talk:Simultaneous orthogonal rotations angle and the associated article history: [1]. Looks like that material was subsequently added here, but it still just duplicates another section so is not needed.--JohnBlackburne^words_deeds 09:26, 19 February 2014 (UTC)

axis-angle to quaternion equation notation

I found the notation in the equation for converting from axis-angle to quaternion confusing. $\mathbf {\omega }$ is used to signify the normalized axis of rotation. However, since $\mathbf {w}$ is used in quaternions, this is confusing. Shouldn't $\mathbf {\hat {e}}$ be used to stay with the notation on this page?

Old equation:

$Q=\left(\cos \left({\frac {\theta }{2}}\right),\omega \sin \left({\frac {\theta }{2}}\right)\right)$

Proposed new equation:

$Q=\left(\cos \left({\frac {\theta }{2}}\right),{\hat {e}}\sin \left({\frac {\theta }{2}}\right)\right)$

BAxelrod (talk) 17:41, 16 September 2013 (UTC)

new edit

Original lead image. A visualization of a rotation represented by an Euler axis and angle.

Hidden image. The orientation of a rigid body indicated with three simultaneous rotations around the rigid body's intrinsic axes and their single rotation equivalent.

Couldn't fit in the edit summary for this edit, so writing it here.

• made consistent the symbols throughout for the first few sections up to Axis–angle representation#Relationship to other representations. (I'll leave the rest alone for now, if omega is used more in this context, if there is agreement I think we should just use e all the way)
• it is not essential to show the Cartesian axes (the definition does not require it), so replaced the lead image with a simpler one containing only the essentials,
• deleted the hidden picture in the source which looks nice but is not specific to this article

M∧Ŝc²ħεИ_τlk 17:17, 3 October 2015 (UTC)

Lie Algebra and so(3)

In the article Rotation group SO(3), the Lie algebra so(3) is defined as the set of skew-symmetric $I\!R^{3x3}$ matrices. Here, it is defined as the vector space $I\!R^{3}$.

In my opinion, since $\exp :so(3)\to SO(3),[v]_{\times }\in so(3)\to R=\exp([v]_{\times })\in SO(3)$, the Lie algebra so(3) should be defined as in the cited article and not as in this one.

Joan Solà (talk) 23:01, 23 December 2016 (UTC)

It is not clear what you are claiming. This article explains and emphasizes the equivalence between 3-vectors and 3x3 skew-symmetric matrices, and provides a wiki link to the cross product matrix, to boot. What would you have in mind to make the obvious equivalence stronger yet???? Cuzkatzimhut (talk) 01:44, 24 December 2016 (UTC)

Maybe I am wrong, but my argument goes like this. so(3), the Lie algebra of SO(3), is the tangent space (one can look at it as the time derivative, or the velocity space) of SO(3) at the origin, and this gives the set of skew symmetric matrices. This vector space so(3) is indeed equivalent (isomorphic) to R3, since it is a vector space with 3 DoF. But this does not mean that R3 is the Lie algebra, it is just a vector space isomorphic to the Lie algebra. Of course due to this isomorphism one can abuse language and call R3 also the Lie algebra. But I think this is not accurate: R3 is not the derivative of SO(3) at the origin. Derivating 3x3 matrices with respect to time, you will always get a 3x3 matrix, and never a 3-vector.

The same goes for the exponential and logarithmic maps. If v is of R3, then exp(v) is not a rotation matrix (who knows what it is). Only exp(S(v)), with S(v) a skew-symmetric matrix isomorphic to v, can be called a member of SO(3) such that R=exp(S(v)). Again, one can abuse the language and say R = exp(v). Indeed, many people do. But this does not make it proper.

So my claim is basically a terminology claim. I am, however, not 100% convinced that R3 cannot be called properly the Lie algebra of SO(3).

Joan Solà (talk) 21:57, 9 September 2019 (UTC)

@Jsola: ${\mathfrak {so}}(3)$ may be presented in either way. Skew-symmetric 3 × 3 matrices indeed form a three-dimensional vector space, and isomorphism can be done in a way intertwining cross product with the matrix commutator. BTW, didn’t even try to decipher ASCII things pretending to be math notation. Incnis Mrsi (talk) 04:50, 10 September 2019 (UTC)