# Cardan angle (zxz, zxzxz) rotation

On the wikipedia page there is a listing of 12 matrices that can be used to represent a yaw-pitch-roll rotation series (YXZ) as a ZXZ rotation, or an XZX rotation, or an XZY rotation..

1) Should the ZXZ rotation matrix be exactly equal to the YXZ rotation matrix?

2) How are the ZXZ etc rotation matrices derived? How would you go about deriving a zxzxz rotation matrix?

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Looking closely, its pretty clear the YXZ rotation matrix is completely different from the ZXZ rotation matrix. Ok. So how do they produce the same rotation when multiplied by a vector? – bobobobo Oct 7 '11 at 22:58
@Henning Ah, I knew someone would take away my zxzxz tag- – bobobobo Oct 7 '11 at 22:58
They don't seem to be any different now... – Daniel McLaury Oct 7 '11 at 23:03
I might have left it if it were xyzzy, though. – Henning Makholm Oct 7 '11 at 23:04

(1) No -- the point (one of the points) is that they give different results. Therefore simply giving three angles is not sufficient to specify a rotation in space; one must also have agreed which of the 12 conventions one is using. (Each of the conventions can specify all rotation matrices, but they do it with different angle triples).

(2) The display above the table of 12 matrices show the rotation matrices $\mathrm{Rot}(Y,\theta)$, $\mathrm{Rot}(X,\theta)$, and $\mathrm{Rot}(Z,\theta)$. Each entry in the table is then just the worked-out product of the three matrices specified -- for example, ZXZ is the matrix product $\mathrm{Rot}(Z,\theta_1)\mathrm{Rot}(X,\theta_2)\mathrm{Rot}(Z,\theta_3)$.

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Can you "convert" from YXZ to ZXZ? – bobobobo Oct 8 '11 at 1:23
Yes, you can, check out here – bobobobo Oct 8 '11 at 1:26