Closed rational trigonometric curves and surfaces

In CAGD curves are described mostly by means of the combination of control points and basis functions. If we associate weights with basis functions and normalize them by their weighted sum, we obtain another set of basis functions that we call quotient bases. We show some common characteristics of curves defined by such quotient basis functions. Following this approach we specify the rational counterpart of the recently introduced cyclic basis, and provide a ready to use tool for control point based exact description of a class of closed rational trigonometric curves and surfaces. We also present the exact control point based description of some famous curves (Lemniscate of Bernoulli, Zhukovsky airfoil profile) and surfaces (Dupin cyclide and the smooth transition between the Boy surface and the Roman surface of Steiner) to illustrate the usefulness of the proposed tool.