Talk:Simultaneous orthogonal rotations angle

Merge discussion

This vector is the same as the rotation vector, which is one form of the Axis-angle representation of rotation. The articles are quite different but this one I think goes into too much detail on a particular application. It also seems seriously mistaken in a couple of ways. Such vectors are not easy to compose as they can't just be added or multiplied: the easiest way to compose them is convert to quaternions, one of the reasons they are often preferred as a rotation representation. It's also not possible to avoid the problem of rotation non-commutativity: rotations in 3D are non-commutative, something which has to be handled by the representation.--JohnBlackburne^words_deeds 18:45, 12 November 2011 (UTC)

I oppose the merge, because this representation has one important difference from axis-angle: the separate angles are easy to understand physically, and can also be measured using gyroscopes. It is true that SORA does not solve commutativity: adding two sets of 3 angles together will not give a combined rotation. -- Petteri Aimonen (talk) 18:06, 27 November 2011 (UTC)

It seems from the last thread on this page we've established (or at least one editor has agreed with me and no-one has disputed it) that this is the same as rotation vector (and is therefore just as easy to understand). It's also been clarified that this does not deal with non-commutativity, removing that source of confusion. It was suggested that this article be merged with both rotation vector and Axis-angle representation but as they are the same article that means merging with just the one article, as I proposed. Any further thoughts?--JohnBlackburne^words_deeds 03:34, 7 December 2011 (UTC)

I agree: There seem to be no difference between SORA and elements of the tangent space of SO(3) (i.e. rotation vector). However, it is a different interpretation: three simultaneous rotations around the Cartesian axis instead of a rotation around a general axis. I think this second interpretation is valuable and should be mentioned in rotation vector. But I fear that a dedicated SORA article is more confusing than helpful. HaukeStrasdat (talk) 15:24, 8 February 2012 (UTC)

Non-commutativity of rotation

The article here presented by User:Sargenije is misleading. Rotations are non-commutative in all presentations. The assertion to the contrary here is suspect. Though the author has a reference, it may not be reliable. The nice diagram does not justify the content.Rgdboer (talk) 19:50, 13 November 2011 (UTC)

This article does not claim that rotations are commutative. SORA denotes a single rotations expressed with three simultanoues rotations around orthogonal axes combined in a single vector. As simultaneous rotations are not sequential, commutativity is not an issue. Of course, when combining multiple sequential rotations, we can not treat their SORA representations as normal vector because rotations are not commutative. The misleading sentence in the article has been modified. — Preceding unsigned comment added by Sargenije (talk • contribs) 12:15, 23 November 2011 (UTC)

Non-commutativity of rotation (once more)

I cannot find the claim that rotations are comutative anywhere in the article. In contrary, it is explicitely states that, except infinitezimal ones, they are non-comutative.

As I see it, the main issue discussed in the article is the fact, that the axes and angular velocity of a rotation can be simply obtained from angular velocities measured with three orthogonal gyroscopes. — Preceding unsigned comment added by 84.20.233.29 (talk) 23:13, 22 November 2011 (UTC)

Non-commutativity of rotation - Reply

The article does not claim that rotations are commutative. It claims that three orthogonal simultaneous rotations are equivalent to a single rotation around a certain axis and for a certain angle. When angular orientation is represented with such three rotations we avoid the problem of different rotation sequence conventions. The problem of rotation non-commutativity is avoided because we are dealing with simultaneous and not sequential rotations. This of course does not mean that rotations are commutative. — Preceding unsigned comment added by 212.235.190.193 (talk) 11:00, 23 November 2011 (UTC)

There is no way to 'avoid the problem'. Rotations in three dimensions (and higher) are generally non-commutative and so all rotation representations have to handle this. If it doesn't then it's not a proper rotation representation. As noted above this rotation representation is already covered at Axis-angle representation, without the errors or other issues with this article. Unless it's established how this is different it should be merged with that article.--JohnBlackburne^words_deeds 11:13, 23 November 2011 (UTC)

Reading these comments I have a feeling that there is certain misunderstanding among participants.

1. SORA is notation obtained directly from three simultaneus orthogonal rotations.

2. It is only a property of SORA, that its length is equal to rotation angle and that its coincides with the equivalent rotation axis. This fact is proven in the cited reference.

3. Successive rotations are not commutative. It does not depend on the notation. So SORA is not commutative.

4. SORA resembles Euler angles, which are also combined from three rotations, however, as the rotations in SORA are simultaneous, there is no problem with different sequences which occurs with Euler angles.

Saso-lkn (talk) 21:30, 23 November 2011 (UTC)

None of that addresses the main problem with this article: a SORA is the same as a rotation vector. The problem is this article does so poor a job of describing it that it's difficult to be sure how it's defined. But if you look at the mathematical content such as converting to a quaternion the formulae are identical, so the representations must be the same. From this article:

$q=\cos(\phi /2)+\mathbf {v} \sin(\phi /2)$

From Axis-angle representation:

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

The author(s) of the references have just rediscovered the rotation vector, something that's been part of mathematics for at least 100 years.--JohnBlackburne^words_deeds 22:20, 23 November 2011 (UTC)

Yes, SORA is rotation vector which is known in mathematics for a long time. What was not known is, that the components of this vector are angles of three simultaneous orthogonal rotations. So there was no direct way from gyroscope measurement to the rotation vector. Without this knowledge, the only way to obtain the rotation vector (or quaternion or any other representation of spatial orentation) from 3D gyroscope mesuremets was to combine a sequence of very small (aproximation of infinitesimally small) rotations. This fact is very important in practical use and it also give a new meaning to the rotation vector. The name SORA only explicitely express this new meaning.

It is also true that axis and angle representation, rotation vector, and even quaternion representation are very similar and they are still regarded as different notations.

So, if there should be any merging, the merging should be done with rotation vector not with axis and angle representation, or, all three should be merged.

Saso-lkn (talk) 10:01, 24 November 2011 (UTC)

The rotation vector representation (combined vector * angle stored in three scalars) is also closely related to the quaternion logarithm. Given a quaternion Q and rotation vector RV, then SORA=RV=log(Q)/2, because the log of a unit quaternion (i.e. a rotation) is a vector (strictly, a pure quaternion) about the rotation axis and of length equal to twice the rotation angle. In reverse, Q=exp(2*RV). Also, the use of the rotation vector, or quaternion logarithm, as a representation of angular velocity, or part of higher-order equations in kinematics, etc, is well established, including the integration of instantaneous angular velocities or accelerations about multiple axes simultaneously to evaluate the combined angular velocity or orientation. It is also fairly widely known and used that any set of infinitesimal rotations commute (including the oarticular case of three about primary axes), and that a non-infinitesimal rotation is the result of this happening over time, as seen in so-called Euler parameter (not Euler angle) equations. Would it be unfair to describe the SORA representation simply as being equal to the classical RV representation, which in turn is simply a half-length version of the classical quaternion logarithm of the rotation in question, with all its existing uses for rotations, angular velocities and accelerations? Or am I missing something. 86.144.198.169 (talk) 00:04, 24 February 2012 (UTC)