Talk:Rotation matrix/Archive 3

Column vector trap
Rotations are frequently composed and it is convenient for the composition to be represented by the product of rotation matrices. But if a column vector is used as input, then it must stand to the right of the matrix. But then a subsequent rotation must stand to the left. The two matrices are then written in an order opposite to a row vector input arrangement. Non-commutativity of multiplication of matrices corresponds to non-commutative rotations. The suggestion is made that column vectors not be used to describe the action of rotation matrices due to the trap that arises with composition. — Rgdboer (talk) 23:36, 29 March 2018 (UTC)


 * I would oppose any change to the current convention because it follows the usage in elementary texts. Rotation matrices for rotations about the origin in 2-D are commutative so there isn't a problem in two dimensions.  Non-commutativity would also be a problem with matrices acting on row vectors.   Dbfirs  19:43, 30 March 2018 (UTC)

First sentence of Complex Planes section is not very clear
Complex planes in M(2,ℝ)

Since $$\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}^2 \ =\ \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}, $$ the plane of matrices $$\begin{pmatrix}x & y \\ -y & x \end{pmatrix}$$ is isomorphic to the complex number plane ℂ, and the above rotation matrix is a point on its unit circle, which acts on the plane as a rotation of &theta; radians.

What is g in section on 'Complex planes in M(2,ℝ)'?
Can't find any reference to what g is. Confusing. — Preceding unsigned comment added by 192.76.8.68 (talk) 16:25, 21 November 2018 (UTC)
 * The expression for g has been expanded using Euler's formula: $$g \ = \ \cos(\theta) + m \sin(\theta) .$$ — Rgdboer (talk) 00:30, 24 November 2018 (UTC)

The figures would be easier to understand if the axes were labeled.
Also, I'm unclear re. the significance of the white plus sign on a green background and white minus sign on a red background. Dr. Phillip M. Feldman (talk) 23:53, 21 February 2019 (UTC)
 * On each file description page is an image of the right-handed coordinate system used.
 * The 3×3 matrix in the top left corner is a signed permutation matrix, so the non-zero entries are 1 and −1.
 * In the images I chose to show only the signs. But I included a text version of the respective matrix on each description page. Watchduck (quack) 17:21, 24 August 2019 (UTC)

Copying material to wikiversity
I'm working on a course on robotic mechanics and modeling. I'm copying material to the following site: https://en.wikiversity.org/wiki/Robotic_Mechanics_and_Modeling/Kinematics Admazzeo (talk) 17:30, 30 January 2020 (UTC)

Potential clarifications in 2.3.1 Determining the axis
I read "Determining the axis" and it took me a while to understand why $$\mathbf{u}$$ is the axis of rotation (in the non-symmetric case). The derivation seems indirect to me so I think it could be improved.

My understanding: the derivation doesn't directly show that $$(R-I)\mathbf{u}=0$$ when considered in reverse, despite assuming it in the forward direction. We could see it if we saw that $$0=\left(R^T+I\right)(R-I)\mathbf{u}$$ but that we can only meaningfully recover that $$(R-I)\mathbf{u}=0$$ through the inversion of $$\left(R^T+I\right)$$ and pre-multiplication with that inverse. $$\left(R^T+I\right)$$ is not invertible if $$R=-I$$ or $$R$$ is a rotation of $$\pi$$ in which case its nullspace is normal to the axis of rotation. The fact that $$(R-I)\mathbf{u}=0$$ is only itself meaningful when $$R\neq I$$. The union of these excluded cases is the symmetric case that $$R=R^T$$.

I propose that instead of saying this:
 * $$\begin{align}

0 &= R^\mathrm{T} 0 + 0 \\ &= R^\mathrm{T}(R - I) \mathbf{u} + (R - I) \mathbf{u} \\ &= \left(R^\mathrm{T}R - R^\mathrm{T} + R - I\right) \mathbf{u} \\ &= \left(I - R^\mathrm{T} + R - I\right) \mathbf{u} \\ &= \left(R - R^\mathrm{T}\right) \mathbf{u}\,, \end{align}$$ we say this:
 * $$\begin{align}

0 &= (R - I) \mathbf{u} \\ &= \left(R^\mathrm{T} + I\right)(R - I) \mathbf{u} \\ &= \left(R^\mathrm{T}R - R^\mathrm{T} + R - I\right) \mathbf{u} \\ &= \left(I - R^\mathrm{T} + R - I\right) \mathbf{u} \\ &= \left(R - R^\mathrm{T}\right) \mathbf{u}\, \end{align}$$ and mention the sub-cases of symmetry in the discussion.

I think it could also help readers' understanding to describe the process geometrically: rotate any unit vector by an angle $$\theta$$ around the rotation axis and subtract the original vector rotated the opposite amount $$\left(R^T=R^{-1}\right)$$. This will produce a vector normal to the rotation axis with magnitude $$2\sin\theta\sin\gamma$$ with $$\gamma$$ being the angle from the vector to the plane normal to the rotation axis. Equate this process to the cross product of a particular vector $$\mathbf{u}$$ with the original unit vector—$$\mathbf{u}$$ is the rotation axis vector with magnitude $$2\sin\theta$$. It took me a while to realize why $$\|\mathbf{u}\|=2\sin\theta$$ so even if it is obvious to some, it may help others to mention this.

Thoughts? ArborealAnole (talk) 02:16, 5 May 2020 (UTC)

More complex planes
The section "Relationship with the complex plane" gives the impression that the matrices of the given "shape" are the only representation of C in 2x2 real matrices. The recent edit removed mention of the great variety of imaginary units found in the matrix ring, and then the various representations of C and the rotation matrices in these representations. The charge of OR ignors the link to M(2,R) where the subject of "rotation matrices" is considered in more detail. Rgdboer (talk) 18:09, 18 February 2021 (UTC)
 * The study of matrix rings that are isomorphic to the complex numbers is completely out of scope in an article about rotation matrices. This is sufficient for removing the mention of matrix rings that are not the ring of the rotation matrices. Moreover, if you think that the removed content is not WP:OR, you have to provide reliable sources. See WP:Wikipedia is not a reliable source. D.Lazard (talk) 20:46, 18 February 2021 (UTC)

Roll pitch yaw conventions
I'm wondering if the terms roll, pitch, and yaw should be exchanged above the matrices in the General rotations section. After a search on the internet, it seems broadly admitted that roll is rotation around the x axis, pitch around y, and yaw around z. Thecrazydonut (talk) 00:57, 2 April 2021 (UTC)


 * The situation here is actually much worse, the matrix in the General rotations section for "yaw, pitch, roll" is just straight up incorrect. Yaw (about Z) is the FIRST rotation that should occur, so it is the LAST matrix in the sequence being multiplied (thus ensuring that it is the first matrix to multiply the column vector). The order should thus be R_x*R_y*R_z (and not R_z*R_y*R_x, which is what the article is currently claiming). --ERC--SP (talk) 19:45, 17 June 2021 (UTC)