Shape preserving representations for trigonometric polynomial curves Original Research Article

This paper has two main goals. Firstly, we show that the space of trigonometric polynomials Tm = span{1, cost, sint, ..., cosmt, sinmt} is not suitable for those methods of CAGD which use control polygons. It is well-known that the bases with good shape preserving properties are the normalized totally positive bases and we prove here that Tm does not possess normalized totally positive bases. Secondly, we show that the space Cm = span{1, cost, ..., cosmt} is suitable for design purposes using control polygons. In fact, we construct a basis Cm of Cm with optimal shape preserving properties and analyze some aspects for the computation of the corresponding curves.