  An arbitrary rigid transformation in $\mathbf{SE}(3)$ can be separated into
two parts, namely, a translation and a rigid rotation. This technical report
reviews, under a unifying viewpoint, three common alternatives to representing
the rotation part: sets of three (yaw-pitch-roll) Euler angles, orthogonal
rotation matrices from $\mathbf{SO}(3)$ and quaternions. It will be described:
(i) the equivalence between these representations and the formulas for
transforming one to each other (in all cases considering the translational and
rotational parts as a whole), (ii) how to compose poses with poses and poses
with points in each representation and (iii) how the uncertainty of the poses
(when modeled as Gaussian distributions) is affected by these transformations
and compositions. Some brief notes are also given about the Jacobians required
to implement least-squares optimization on manifolds, an very promising
approach in recent engineering literature. The text reflects which MRPT C++
library functions implement each of the described algorithms. All formulas and
their implementation have been thoroughly validated by means of unit testing
and numerical estimation of the Jacobians
