Talk:Euler's rotation theorem

quaternions

Why is this article more about quaternions than it is about Euler rotation? One reference to quaternion algebra is adequate. The topic has been obfuscated by a writer who wants to steer readers. 65.175.151.8 15:08, 24 October 2006 (UTC)Tim Golden BandTechnology.com

they are intimately connected, same thing in different disguises. IMHO, the section on quaternions could use some elaboration and clarification. Mct mht 01:42, 8 February 2007 (UTC)

unclear about number of parameters and quaternion significance

• Technically one needs only 3 numbers to represent an arbitrary rotation, since the rotation axis can be normalized, with its magnitude representing the angle (I suspect that this closer to what Euler's rotation theorem states)

yes, you're right. maybe that'll get corrected. Mct mht 01:42, 8 February 2007 (UTC)

• The article makes it sound as if any set of 4 numbers representing a rotation is called a quaternion. This is untrue, since a quaternion has a specific mathematical meaning, and is used to represent 3D rotation in a specific way.

right there also. current version can certainly be improved. rotations corresponds to unit quaternions, in a "1 to 2" way, meaning unit quaternions is a double cover of SO(3). Mct mht 01:42, 8 February 2007 (UTC)

Show calculation of axis and angle of rotation

Should we show how to calculate the angle and axis of rotation? I have the formula here:

As a corollary of the theorem follows that the trace of a rotation matrix (R) is equal to:

$Trace[R]=2*cos(\theta )+1$

so s can be written as:

$s=\arccos {\left[{\frac {Trace(R)-1}{2}}\right]}=\arccos {\left[{\frac {r_{00}+r_{11}+r_{22}-1}{2}}\right]}$

Second, the unique axis of rotation is defined as:

${\vec {k}}={\frac {1}{2\sin {(\theta )}}}*\left({\begin{matrix}r_{21}-r_{12}\\r_{02}-r_{20}\\r_{10}-r_{01}\end{matrix}}\right)$

and the magnitude:

$|{\vec {k}}|={\frac {1}{2\sin {(\theta )}}}*{\sqrt {(r_{21}-r_{12})^{2}+(r_{02}-r_{20})^{2}+(r_{10}-r_{01})^{2}}}$

BUT I need a citation for these values...

Excursion into matrix theory

The statement

An m×m matrix A has m orthogonal eigenvectors if and only if A is normal.

is false. The statement should read

An m×m matrix A can be diagonalised if and only if A is normal. An orthogonal set of m eigenvectors can be obtained if A is hermitian. The eigenvectors of a rotation matrix are not in general mutually orthogonal.

Really? "a matrix is normal if and only if its eigenspaces span Cⁿ and are pairwise orthogonal with respect to the standard inner product of Cⁿ" (Normal matrix#Consequences). Boris Tsirelson (talk) 10:33, 19 April 2010 (UTC)

The matrix proof is wrong

The proof shows that a rotation matrix has an invariant vector. Period. It does not show that "in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis". When saying "An algebraic proof starts from the fact that a rotation is a linear map in one-to-one correspondence withi a 3×3 rotation matrix R", you are already assuming what you want to show. That is, that the cartesian coordinates are related by a rotation about an axis. --190.188.3.11 (talk) 16:22, 23 June 2010 (UTC)

Two cartesian coordinates are related by a rotation matrix whose columns are the vectors of the second frame. The matrix has a real eigenvector v and is linear, therefore any vector k.v is also a eigenvector. This eigenspace is the rotation axis.--94.126.240.2 (talk) 07:36, 25 June 2010 (UTC)

You say that "Two cartesian coordinates are related by a rotation matrix whose columns are the vectors of the second frame." It seems this should be mentioned in the article, and proved too. Although it may seem obvious that an isometry S such that S (0) = 0 is linear, this has to be shown. So it seems that several things are being assumed because they are "obvious", making the proof not rigorous.--190.188.3.11 (talk) 04:12, 2 July 2010 (UTC)

I can't see any problems with it. The only thing it asserts is that the rotation can be represented by a matrix with the property

$\mathbf {R} ^{\mathrm {T} }\mathbf {R} =\mathbf {R} \mathbf {R} ^{\mathrm {T} }=\mathbf {E}$,

i.e. its inverse is its transpose. This is non-obvious but is easy enough to show by e.g. building the matrix up from rotations about three orthogonal axes. All rotations can be generated this way, otherwise gimbals would not work. Other than that the only thing established is that the determinant is +1 and not -1, but that follows from considering the volume-preserving nature of rotations. From these properties the unit root is found algebraically, and the existence of a fixed vector deduced from the theory of eigenvectors. --JohnBlackburne^words_deeds 11:41, 2 July 2010 (UTC)

I do nothave all these things very clear, but the article does not define what a rotation is. What is a rotation? The algebraic proof shows that a rotation matrix has an eigenvector; it does not show that two cartesian coordinates are related by a rotation. Reflextions don´t also preserve volume? As far as I know, that the determinant is 1 is shown using continuity arguments.

--190.188.3.11 (talk) 20:22, 7 August 2010 (UTC)