A Circle-Preserving Variant of the Four-Point Subdivision Scheme

The four-point curve subdivision scheme is one of the classic reference points of subdivision theory. It has effective C2 continuity, although the curvature at the data points actually diverges slowly to infinity as very large numbers of subdivision steps are taken. However, it has rather large longitudinal artifacts, so that points interpolated around a curve of almost constant curvature are fitted by a curve with significant variations of curvature. We describe here a geometry-sensitive variant of this scheme which does not have this problem. In fact circles are reproduced exactly with any spacing of the initial data. §1. The Four-Point Scheme The four point scheme is a uniform stationary subdivision scheme with the mask [−1, 0, 9, 16, 9, 0,−1]/16. It was first described by Dyn, Levin and Gregory in [3], although the functional version had already been described by Dubuc in [1]. There is a new vertex at each old vertex, and also a new vertex associated with each edge of the control polygon, and these new vertices are given by the stencil [−1, 9, 9,−1]/16. Thus, in Figure 1, 16P = 9[B + C] − [A + D], applying to each coordinate independently. It is well-known [2] that this scheme has a limit curve which is almost C2 (“almost” is due to a Jordan block in the eigenanalysis at the dyadic points, so that the second divided differences there increase at each subdivision step by a fixed amount proportional to the original fourth differences), but that the shape is somewhat prone to longitudinal artifacts (the ripples which occur at one cycle per data point, visible primarily in the curvature plots). The magnitude of this artifact (tabulated in Sect. 9) is about three times as large as that of the cubic B-Spline, and this scheme Mathematical Methods for Curves and Surfaces: Tromsø 2004 1 M. Dæhlen, K. Mørken, and L. L. Schumaker (eds.), pp. 1–12. Copyright oc 2005 by Nashboro Press, Brentwood, TN. ISBN 0-9728482-4-X All rights of reproduction in any form reserved. 2 M. Sabin and N. Dodgson