Geometric conditions for tangent continuity of interpolatory planar subdivision curves

Abstract

Curve subdivision is a technique for generating smooth curves from initial control polygons by repeated refinement. The most common subdivision schemes are based on linear refinement rules, which are applied separately to each coordinate of the control points, and the analysis of these schemes is well understood. Since the resulting limit curves are not sufficiently sensitive to the geometry of the control polygons, there is a need for geometric subdivision schemes. Such schemes take the geometry of the control polygons into account by using non-linear refinement rules and are known to generate limit curves with less artefacts. Yet, only few tools exist for their analysis, because the non-linear setting is more complicated. In this paper, we derive sufficient conditions for a convergent interpolatory planar subdivision scheme to produce tangent continuous limit curves. These conditions as well as the proofs are purely geometric and do not rely on any parameterization.