  Triple orthogonal coordinate systems having coordinate lines as circles or
straight lines are considered. Technically, they are represented by trilinear
rational quaternionic maps and are called Dupin cyclidic cubes, naturally
generalizing the bilinear rational quaternionic parametrizations of principal
patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are
studied and classified up to M\"obius equivalency in Euclidean space.
