Conversion between quaternions and Euler angles

The orthogonal matrix corresponding to a rotation by the unit quaternion z = a + bi + cj + dk (with |z| = 1) is given by

${\begin{bmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2bc-2ad&2ac+2bd\\2ad+2bc&a^{2}-b^{2}+c^{2}-d^{2}&2cd-2ab\\2bd-2ac&2ab+2cd&a^{2}-b^{2}-c^{2}+d^{2}\\\end{bmatrix}}$

The orthogonal matrix corresponding to a rotation with Euler angles $\phi (roll),\theta (pitch)and\psi (yaw)$, is given by

${\begin{bmatrix}cos\theta cos\psi &-cos\phi sin\psi +sin\phi sin\theta cos\psi &sin\phi sin\psi +cos\phi sin\theta cos\psi \\cos\theta sin\psi &cos\phi cos\psi +cos\phi sin\theta sin\psi &-sin\phi cos\psi +cos\phi sin\theta sin\psi \\-sin\theta &sin\phi cos\theta &cos\phi cos\theta \\\end{bmatrix}}$