A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^1$$\end{document}, and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^2$$\end{document} construction and various possible extensions of the method.