We introduce an alternative way of constructing continuous flexible tubes and tubular structures based on a discrete, semi-discrete and smooth construction of surfaces known as T-hedra in the discrete case and profile-affine surfaces in the smooth setting, respectively. The geometric understanding of this method enables us to generalize discrete tubes with a rigid-foldability and to extend the construction to smooth and semi-discrete tubes with an isometric deformation.This achievement implies a unified treatment of continuous flexible structures, like surfaces and metamaterials composed of tubes, and it is the base for a deeper study of zipper tubes and their generalization.