  This paper proposes a robust algorithmic and computational framework to
address the problem of modeling the volume obtained by sweeping a solid along a
trajectory of rigid motions. The boundary representation (simply brep) of the
input solid naturally induces a brep of the swept volume. We show that it is
locally similar to the input brep and this serves as the basis of the
framework. All the same, it admits several intricacies: (i) geometric, in terms
of parametrizations and, (ii) topological, in terms of orientations. We provide
a novel analysis for their resolution. More specifically, we prove a
non-trivial lifting theorem which allows to locally orient the output using the
orientation of the input. We illustrate the framework by providing many
examples from a pilot implementation.
