Generalized cylinders (GCs) are a popular representational tool in computer vision. In medical imaging, the curved axis GC is particularly applicable to a number of elongated physical structures such as vasculature, bone and bronchi. In many of these instances, it is necessary to recover curved-axis GCs with arbitrary cross-sections. It is also vital that these structures, once recovered, can be analyzed and visualized with off-the-shelf algorithms and software packages. Such tools are usually designed to operate on the domain of polynomial or rational surfaces. Unfortunately most extant, suitably versatile GC representations do not admit rational parameterizations. We develop an entirely rational B-spline representation for generalized cylinders with curved axes and arbitrary cross-section functions. We demonstrate how our representation can be used as a deformable model by extracting a rational GC from pre-segmented spinal data using a discrete, dynamic surface fit.