Roto-translation (rigid transformation)

The rigid transformation (or roto-translation motion) is the composition between reflection, translation, and rotation, and therefore it is an isometry, that is, a geometric transformation that leaves the distances unchanged. In other words, we can think of roto-translation as a rigid movement in which a geometric figure first rotates and then translates.

The rototranslation motion of a rigid body has 6 degrees of freedom (3 of translation and 3 of rotation). To define a rototranslation we need:

  • a point in the plane or a straight line in space with respect to which rotate;
  • an angle \(\alpha\) characterized by an amplitude and a direction (clockwise or counterclockwise);
  • a vector \(\vec{v}\) against which to translate.

It is important to respect the order in which the two transformations are carried out: first rotate and then translate. If the order is reversed and is executed before the translation and then the rotation, it could happen to get with a figure that has a different position than it should have.