We propose a new parametric 3D snake with cylindrical topology. Its construction is based on interpolatory basis functions which facilitates user-interaction because the control points of the snake directly lie on the surface of the deformable cylinder. We prove that the basis functions exactly reproduce a cylinder and propose a new parametrization as a tensor-product spline surface. We provide explicit formulas for the energy function based on Green's theorem that speed up the computation of the optimization algorithm. We have implemented the proposed framework as a freely available open-source plugin for the bioimaging platform Icy. Its utility has been tested on phantom data as well as on real 3D data to segment the spinal cord and the descending aorta.