In a previous paper (Lipovetsky and Dyn in Comput Aided Geom Des 48:36–48, 2016), we introduced a weighted binary average of two 2D point-normal pairs, termed circle average, and investigated subdivision schemes based on it. These schemes refine point-normal pairs in 2D and converge to limit curves and limit normals. Such a scheme has the disadvantage that the limit normals are not the normals of the limit curve. In this paper, we address this problem by proposing a new averaging method and obtaining a new family of algorithms based on it. We demonstrate their new editing capabilities and apply this subdivision technique to smooth a precomputed feasible polygonal point robot path.

在先前的论文（Lipovetsky和Dyn在Comput Aided Geom Des 48：36–48，2016）中，我们引入了两个二维点-法线对的加权二进制平均值，称为圆平均值，并基于此对细分方案进行了研究。这些方案在2D中精炼了点法线对，并收敛到极限曲线和法线。这种方案的缺点是极限法线不是极限曲线的法线。在本文中，我们通过提出一种新的平均方法并基于该方法获得新的算法家族来解决此问题。我们展示了他们的新编辑功能，并应用了这种细分技术来平滑预先计算的可行多边形点机器人路径。