Control point based exact description of a class of closed curves and surfaces

Based on cyclic curves/surfaces introduced in Róth et al. (2009), we specify control point configurations that result an exact description of those closed curves and surfaces the coordinate functions of which are (separable) trigonometric polynomials of finite degree. This class of curves/surfaces comprises several famous closed curves like ellipses, epi- and hypocycloids, Lissajous curves, torus knots, foliums; and surfaces such as sphere, torus and other surfaces of revolution, and even special surfaces like the non-orientable Roman surface of Steiner. Moreover, we show that higher order (mixed partial) derivatives of cyclic curves/surfaces are also cyclic curves/surfaces, and we describe the connection between the cyclic and Fourier bases of the vector space of trigonometric polynomials of finite degree.