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Any proper rotation (in three dimensions) can be expressed using the Tait-Bryan (sometimes called improper Euler) angles in the form $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_x(\psi) $$ where $R_z(\phi)$ is the a rotation by $\phi$ about the Z axis, and similarly for the other terms.

Let's define a conjugate rotation $R^C$ by the rotation expressed using the negatives of those same Euler angles: $$ [R(\phi,\theta,\psi)]^C=R_z(-\phi)R_y(-\theta)R_x(-\psi) $$ Now let's define an "undone rotation" as the result of somebody trying to undo a rotation $R$ by multiplying $R^C$. That is, $$ U(\phi,\theta,\psi) = R_z(\phi)R_y(\theta)R_x(\psi)R_z(-\phi)R_y(-\theta)R_x(-\psi) $$ The set $\left\{U\right\}$ of all undone rotations (defined in this way) is labeled by three real parameters, and all members are proper rotations, but it is by no means clear that every rotation can be expressed as a member of $\left\{U\right\}$. In fact, but I have not been able to find a set of $(\phi,\theta,\psi)$ such that $U(\phi,\theta,\psi) = R_x(\alpha)$ for arbitrary values of $\alpha$.

My question is, does the set $\left\{U\right\}$ cover the entire rotation group? If not, is there an easy way to characterize those rotations which are or are not a member of $\left\{U\right\}$? And if $\left\{U\right\}$ is not the whole rotation group, is it a subgroup of it?

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  • $\begingroup$ Maybe this is just a silly question, but why would the composition of two elements of $\{U\}$ be in $\{U\}$? It doesn't looked closed under composition. $\endgroup$ – Robin Goodfellow Sep 19 '14 at 2:10
  • $\begingroup$ It may or may not be. I haven't been able to find a rotation that I could prove is not in $\left\{U\right\}$. If $\left\{U\right\}$ is not closed under composition, then of course it does not form a group, and it does not cover the rotation group, in which case the relevant question is whether there is an easy way to characterize those values of $(\phi,\theta,\psi)$ for which $R(\phi,\theta,\psi) \not \in \left\{U\right\}$. $\endgroup$ – Mark Fischler Sep 19 '14 at 14:52
  • $\begingroup$ As an aside, the set of rotations expressible as $R^2$ for some rotation $R$ does cover the whole rotation group. $\endgroup$ – Mark Fischler Sep 20 '14 at 14:10
  • $\begingroup$ I have been told that these undone rotatoins can't cover all the rotations because how would you get a non-trivial pure-X rotation. But empirically,$R_z(23.605)R_y(23.605)R_x(5)R_z(-23.605)R_y(-23.605)R_x(-5) = R_x(-10)$ to 4 places of accuracy (all angles here are in degrees). $\endgroup$ – Mark Fischler Oct 3 '14 at 15:57
  • $\begingroup$ In fact, arbitray rotations about any of the 3 axes (x, y, or z) can be obtained. For an X rotation use the pattern above; for a Z rotation do something similar: $R_z(10)R_y(32.96)R_x(-32.96)R_z(-10)R_y(-32.96)R_x(32.96)=R_z(20)$. And for a Y rotation, merely use a 180 degree Z rotation, a Y rotation about the half angle, and no X rotation. $\endgroup$ – Mark Fischler Oct 3 '14 at 19:19

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