Kim and Moon (2017) have recently proposed rectifying control polygons as an alternative to Bézier control polygons and a way of controlling planar PH curves by the rectifying control polygons. While a Bézier control polygon determines a unique polynomial curve, a rectifying control polygon gives a multitude of PH curves. This multiplicity of PH curves naturally raises the selection problem of the “best” PH curves, which is the main topic of this paper.

To resolve the problem, we first classify PH curves of degree $2n+1$ into $2^n$ subclasses by defining the types of PH curves, and propose the absolute hodograph winding number as a topological index to characterize the topological behavior of PH curves in shape. We present a lower bound of the topological index of a PH curve which is given solely by its type, and prove the uniqueness of the best PH curve by exploiting it. The existence theorems are also proved for cubic and quintic PH curves. Finally, we propose a selection rule of the best PH curve only based on its type.

Kim和Moon（2017）最近提出了对控制多边形进行精整的建议，以替代Bézier控制多边形，并提出了通过对控制多边形进行精整来控制平面PH曲线的方法。当贝塞尔（Bézier）控制多边形确定唯一的多项式曲线时，整流控制多边形会给出许多PH曲线。PH曲线的多样性自然引起了“最佳” PH曲线的选择问题，这是本文的主题。

为了解决这个问题，我们首先对度数的PH曲线进行分类 $2ñ+\mathrm{1个}$ 进入 $2^{ñ}$通过定义PH曲线的类型来划分子类，并提出绝对全息图绕组数作为拓扑指标，以表征形状中的PH曲线的拓扑行为。我们给出了仅由其类型给出的PH曲线拓扑指数的下限，并通过利用它来证明最佳PH曲线的唯一性。还证明了三次和五次PH曲线的存在性定理。最后，我们仅根据其类型提出最佳PH曲线的选择规则。