Shape control tools for periodic Bézier curves

Abstract

Bézier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flexibility, and thus more diverse types of curves, can be achieved by associating a weight with each control point, that is, by considering rational Bézier curves. As shown by Ramanantoanina and Hormann (2021), additional and more direct control over the curve shape can be achieved by exploiting the correspondence between the rational Bézier and the interpolating barycentric form and by exploring the effect of changing the degrees of freedom of the latter (interpolation points, weights, and nodes). In this paper, we explore similar editing possibilities for closed curves, in particular for the rational extension of the periodic Bézier curves that were introduced by Sánchez-Reyes (2009). We show how to convert back and forth between the periodic rational Bézier and the interpolating trigonometric barycentric form, derive a necessary condition to avoid poles of a trigonometric rational interpolant, and devise a general framework to perform degree elevation of periodic rational Bézier curves. We further discuss the editing possibilities given by the trigonometric barycentric form.