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.

贝塞尔曲线是曲线设计的必备工具。由于它们的属性，可以通过简单地修改曲线的控制多边形来将平移、旋转或缩放等常见操作应用于曲线。通过将权重与每个控制点相关联，即通过考虑有理贝塞尔曲线，可以实现更大的灵活性，从而实现更多样的曲线类型。如Ramanantoanina 和 Hormann (2021)所示，通过利用有理贝塞尔曲线和插值重心形式之间的对应关系，并通过探索改变后者的自由度（插值点、权重和节点）的影响，可以实现对曲线形状的额外和更直接的控制。在本文中，我们探索了闭合曲线的类似编辑可能性，特别是Sánchez-Reyes (2009)引入的周期性贝塞尔曲线的合理扩展. 我们展示了如何在周期性有理贝塞尔曲线和插值三角重心形式之间来回转换，推导避免三角有理插值极点的必要条件，并设计一个通用框架来执行周期性有理贝塞尔曲线的度数提升。我们进一步讨论三角重心形式给出的编辑可能性。