ABSTRACT
Rigid motions are at the heart of Euclidean geometry. The rigid motions consist of composites of rotations, translations, and reflections. Thus, for the dual quaternions representing translation Furthermore, for dual quaternions representing rotations, the coefficient of is zero, so Therefore, the authors have the compatible sandwiching formulas for translation and rotation. But, so they can generalize Theorem still further to screw transformations about arbitrary lines L by composing a rotation around the line L with a translation in the direction parallel to the line L. Hence The screw transformations contain both the translations and the rotations as special cases. Since the product of rotations about the origin is a single rotation about the origin and the product of translations is another translation, every rigid motion is equivalent to one rotation about the origin followed by one translation.
