Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.