Expressing a HepRotation as HepAxisAngle or HepEulerAngles

It is worth noting that equations () and () are not single-valued when extracting $( \hat{u}, \delta)$ or $(\phi, \theta, \psi)$ from a Rotation matrix. We adhere to the following conventions to resolve that ambiguity whenever forming an HepAxisAngle or HepEulerAngles from a Rotation$R$:

$$R = {\bf I} \Longrightarrow \delta = 0, \; \hat{u} = \hat{z}$$ (142)

$$R \longrightarrow ( \hat{u}, \delta ) \Longrightarrow 0 \leq \delta \leq \pi$$ (143)

$$R \longrightarrow ( \phi, \theta, \psi ) \Longrightarrow 0 \leq \theta \leq \pi$$ (144)

$$R \longrightarrow ( \phi, \; \theta = 0, \; \psi ) \Longrightarrow -\pi/2 < \phi = \psi \leq \pi/2$$ (145)

$$R \longrightarrow ( \phi, \; \theta = \pi, \; \psi ) \Longrightarrow -\pi/2 < \phi = -\psi \leq \pi/2$$ (146)

Thus when supplying values for Euler angles, we return $(0, 0, 0)$ whenever the rotation is equivalent to the identity, and return $(0, \pi, 0)$ in preference to the equivalent $(\pi, \pi, \pi)$.

In particular, special case rotations such as HepRotationX with $\delta$ supplied as zero or $\pi$ obey these conventions: even though for $\delta = \pi - \vert\epsilon\vert$ a RotationX would have Euler angles $( \pi, \delta, \pi, )$, when $\delta = \pi$ exactly, the Euler angles are $(0, \delta=\pi, 0)$.

These conventions for how methods return values for Euler angles do not affect the user's right to supply explicit values for $( \hat{u}, \delta)$ or $(\phi, \theta, \psi)$ when defining one of those structures. Similarly, the conventions for reading axis and angle do not affect the user's ability to supply arbitrary values for the angle $\delta$.

Obeying the above rules, Euler angles returned for rotations about coordinate axes behave as follows:

$$\mbox{HepRotationX}(\delta) \longrightarrow \left\{ \begin{a... ... ( 0, \pi, 0 ) & \mbox{ if } & \delta = \pi \end{array}\right.$$ (147)

$$\mbox{HepRotationY}(\delta) \longrightarrow \left\{ \begin{a... ...elta, -\pi/2 ) & \mbox{ if } & \delta = \pi \end{array}\right.$$ (148)

$$\mbox{HepRotationZ}(\delta) \longrightarrow ( -\delta/2, 0, -\delta/2 )$$ (149)

always assuming that $\delta$ represents an angle of active rotation between $\-pi$ and $\pi$.