> ...describing sofas in terms of the translations and rotations as a function of time is by now a standard way of thinking about the moving sofa problem, and has been used in the attack on the problem by Joseph Gerver in his paper from 1992, and in my own recent paper, where among other thing I made precise your suggested notion of describing the sofa shape that results from the continuous intersection of corridors rotating and translating over time. See my paper (available here: https://www.math.ucdavis.edu/~romik/data/uploads/papers/sofa...), and especially Theorem 1 in section 2.

Dan Romik