Computer Aided Geometric Design 27 (2010) 78–95 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Variations on the four-point subdivision scheme U.H. Augsdörfer a,∗ , N.A. Dodgson a , M.A. Sabin b a Computer Laboratory, University of Cambridge, 15 J.J. Thomson Ave, Cambridge CB3 OFD, England, United Kingdom b Numerical Geometry Ltd., 19 John Amner Close, Ely, Cambridge CB6 1DT, England, United Kingdom article info abstract Article history: Received 29 August 2008 Received in revised form 25 September 2009 Accepted 25 September 2009 Available online 30 September 2009 Keywords: Subdivision Four-point scheme Curvature Geometric sensitivity Circle preserving A step of subdivision can be considered to be a sequence of simple, highly local stages. By manipulating the stages of a subdivision step we can create families of schemes, each designed to meet different requirements. We postulate that such modification can lead to improved behaviour. We demonstrate this using the four-point scheme as an example. We explain how it can be broken into stages and how these stages can be manipulated in various ways. Six variants that all improve on the quality of the limit curve are presented and analysed. We present schemes which perfectly preserve circles, schemes which improve the Hölder continuity, and schemes which relax the interpolating property to achieve higher smoothness. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Given a sequence of vertices, subdivision is a process by which, in each refinement step, new vertices are inserted as linear combinations of old vertices. Repeating the process leads eventually to a smooth limit curve. We can build each refinement step out of a sequence of simple, local stages. This idea is expressed, for example, in the Lane–Riesenfeld subdivision construction for B-spline curves (Lane and Riesenfeld, 1980) in which, after a single duplication stage in which the number of control points is doubled by just taking each point twice, a sequence of smoothing operators is applied. The same motif is present in the original description of Catmull-Clark surface subdivision, where each refinement is expressed in three stages (Catmull and Clark, 1978). Zorin and Schröder (2001) and Oswald and Schröder (2003) use this idea to generate families of subdivision schemes with large support by varying the number of smoothing stages. We show that we can produce families of subdivision schemes by manipulating, in various ways, the individual stages within a subdivision step. This has advantages when designing subdivision schemes for particular purposes. Our examples show that such modification can lead to improved behaviour. This idea is illustrated using the four-point subdivision scheme (Dyn et al., 1987). We first describe the original four- point subdivision scheme and then six variations on the scheme which are obtained by tuning the local stages in various ways, producing some interesting subdivision schemes all of which are improvements on the original. We present schemes which perfectly preserve circles, schemes which improve the Hölder continuity, and schemes which relax the interpolating property to achieve even higher smoothness. We analyse these schemes and discuss their relative merits. * Corresponding author. E-mail addresses: Ursula.Augsdorfer@cl.cam.ac.uk (U.H. Augsdörfer), Neil.Dodgson@cl.cam.ac.uk (N.A. Dodgson), malcolm@geometry.demon.co.uk (M.A. Sabin). 0167-8396/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2009.09.002