Shape preserving representations for trigonometric polynomial curves

Abstract

This paper has two main goals. Firstly, we show that the space of trigonometric polynomials T m = span{1, cos t, sin t, ..., cos mt, sin mt} 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 T m does not possess normalized totally positive bases. Secondly, we show that the space C m = span{1, cos t, ..., cos mt} is suitable for design purposes using control polygons. In fact, we construct a basis C m of C m with optimal shape preserving properties and analyze some aspects for the computation of the corresponding curves.