#### Two screws

The homogeneous transformation matrix will be constructed by combining two simpler transformations. The transformation

 (3.54)

causes a rotation of about the -axis, and a translation of along the -axis. Notice that the rotation by and translation by commute because both operations occur with respect to the same axis, . The combined operation of a translation and rotation with respect to the same axis is referred to as a screw (as in the motion of a screw through a nut). The effect of can thus be considered as a screw about the -axis. The second transformation is

 (3.55)

which can be considered as a screw about the -axis. A rotation of about the -axis and a translation of are performed.

Steven M LaValle 2012-04-20