Control point based exact description of a class of closed curves and surfaces
The authors consider closed parametric curves and surfaces based on cyclic basis functions which were introduced by \textit{Á. Róth}, \textit{I. Juhász}, \textit{J. Schicho} and \textit{M. Hoffmann} [Comput. Aided Geom. Des. 26, No. 5, 528--546 (2009; Zbl 1205.65345)] as an alternative for say \(B\)-spline functions. Cyclic basis functions are trigonometric polynomials and thus allow for exact representations of a wide class of famous closed curves and surfaces. In this paper explicit formulas for the control points (coefficients of the representations) are given and illustrated by several examples including epi- and hypocycloids, Lissajous curves, torus knots, or surfaces of revolution, as well as the Roman surface of Steiner. Moreover, the authors describe by explicit formulas the transformation between the basis of cyclic functions and the standard basis of trigonometric polynomials, and show that partial derivatives of cyclic curves/surfaces are cyclic curves/surfaces.
scientific article; zbMATH DE number 5869102
scientific article