Computer Aided Geometric Design 28 (2011) 198–211 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra ✩ U.H. Augsdörfer a,∗ , N.A. Dodgson b , M.A. Sabin c a Institute of Computer Graphic and Knowledge Visualisation, TU Graz, Inffeldgasse 16c, 8010 Graz, Austria b The Computer Laboratory, University of Cambridge, 15 J.J. Thomson Ave, Cambridge CB3 OFD, England, United Kingdom c Numerical Geometry Ltd., 19 John Amner Close, Ely, Cambridge CB6 1DT, England, United Kingdom article info abstract Article history: Available online 19 February 2011 Keywords: Subdivision Artifact Loop subdivision Butterfly subdivision Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra (Sabin et al., 2005; Augsdörfer et al., 2011). In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Subdivision is an algorithmic technique to generate smooth surfaces as the limit of a sequence of successively refined polygons or polyhedra. The limit surface is made up of components with spatial frequencies below the Nyquist limit and those with spatial frequencies above. The first group of components can be controlled by the designer by moving control points and represents the desired surface. The second group cannot be controlled by the designer. We call these components artifacts. For all schemes these artifacts are visible as peaks per sampling point in a curvature plot (Sabin et al., 2005). We can decrease, but not eliminate, the size of curvature variation by increasing the density of the control points, or by choosing a subdivision scheme based on a B-spline of higher degree. Sabin et al. (2005) explained how subdivision can be employed as a tool for analysis of artifacts which are not control- lable by the designer. Augsdörfer et al. (2011) employed this idea to analyse the limit surfaces of various algorithms based on quadrilateral meshes. In this work we develop a generic expression based on the tools described by Sabin et al. (2005) which is applicable to limit surfaces irrespective of the mesh type. We demonstrate this method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes. The three examples analysed are limit surfaces obtained using Loop subdivision (Loop, 1987), Butterfly subdivision (Dyn et al., 1990) and a novel interpolating subdivision scheme with two smoothing stages. We compare our results for these limit surfaces to results obtained for surfaces defined by algorithms based on quadrilateral meshes. ✩ This paper has been recommended for acceptance by H. Prautzsch. * Corresponding author. E-mail addresses: Ursula.Augsdorfer@cgv.tugraz.at (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 © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2011.01.003