$C^1$ analysis of 2D subdivision schemes refining point-normal pairs with the circle average

TL;DR

This paper proves that certain 2D subdivision schemes refining point-normal pairs, based on the circle average, produce curves with continuous first derivatives, extending previous convergence results.

Contribution

It demonstrates that the curves generated by modified Lane-Riesenfeld and 4-Point schemes are $C^1$, advancing understanding of their smoothness properties.

Findings

The schemes produce $C^1$ continuous curves.

Convergence of the schemes is established.

Extension of previous convergence results to smoothness.

Abstract

This article continues the investigation started in [9] on subdivision schemes refining 2D point-normal pairs, obtained by modifying linear subdivision schemes using the circle average. While in [9] the convergence of the Modified Lane-Riesenfeld algorithm and the Modified 4-Point schemes is proved, here we show that the curves generated by these two schemes are $C^1$.