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This is what Wikipedia says:

\begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{bmatrix}

This is what I think it should be:

\begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \\ \end{bmatrix}

Which is correct?

The convention used on the Wikipedia page for the other two transformations suggests that they are using a different standard for rotations about the x axis and the z axis than they are for rotations about the y axis.

theo-brown
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okarin
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  • Wikipedia is fine. Let us see: Put the x-axis on the floor, pointing to your right, the y-axis pointing up from the floor towards you. Then the z-axis, following the rule of the right-hand, which is used as a way to agree on a positive orientation, should be pointing to your back. So, to achieve a counterclockwise rotation, as the angle increases from zero, the z-coordinate should go to negatives first. Try rotating [1,0,0]. – OR. Dec 05 '13 at 18:32
  • You forgot to mention whether you're multiplying these matrices with row vectors or column vectors. – J. M. ain't a mathematician Dec 28 '16 at 11:06

2 Answers2

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Talking about 'matters of opinion' is misleading. It's not a matter of opinion - it's a matter of mathematical convention due to the cyclic right-handed nature of the xyz co-ordinates.

See: Why is rotation about the y axis in $\mathbb{R^3}$ different from rotation about the x and y axis.

theo-brown
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It's, for all practical purposes, a matter of opinion in this case. It depends on where you are measuring $\theta$ from, and different people have different conventions in $3$-space.

Tim Ratigan
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  • Here is where it is in Wikipedia: https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations The convention they use for the other two transformations indicates that they are using a different standard for rotations about the x axis and the z axis than they are for rotations about the y axis. – okarin Dec 05 '13 at 18:23
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    Their rotation matrix measures $\theta$ as clockwise rotation in the $x-z$ plane if $z$ takes the place of the $y$-axis in the conventional Euclidean plane. However, if you switch the axes labels, their convention makes sense. Your matrix is more more widely accepted, but theirs isn't strictly wrong. – Tim Ratigan Dec 05 '13 at 18:28
  • That is not really the case. If the y-axis is pointing towards you, as wikipedia is asking, then the z-axis should be pointing down so that it agrees with the positive orientation of the coordinate system. – OR. Dec 05 '13 at 18:33
  • @ABC at which point? I'm just speaking from what I've seen, I could be wrong about accepted convention. – Tim Ratigan Dec 05 '13 at 18:36
  • As a convention angles are measured positive counterclockwise from the positive part of the x-axis, while the axis of rotation is pointing up towards you, from the plane of rotation. In addition the positive orientation of the axes should satisfy the rule of the right hand. Then if the x-axis is to the right, and y-axis is up from the plane, then the z-axis should be pointing down (or rather back, like the negative part of the usual y-axis in a plane). – OR. Dec 05 '13 at 18:40
  • The z-axis is normally up from the plane though. – okarin Dec 05 '13 at 23:59