We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Durham University

Research & business

View Profile

Publication details for Dr Ioannis Ivrissimtzis

Ivrissimtzis, Ioannis, Dodgson, Neil & Sabin, Malcolm (2004). A generative classification of mesh refinement rules with lattice transformations. Computer Aided Geometric Design 21(1): 99-109.

Author(s) from Durham


We give a classification of subdivision refinement rules using similarity transformations of lattices. Our work expands recent results in the classification of primal triangular subdivision. In the examples we concentrate on the cases with a low ratio of similarity and find new univariate and bivariate refinement rules with the lowest possible such ratio, showing that this very low ratio usually comes at the expense of symmetry.


Akleman, E., Srinivasan, V., 2002. Honeycomb subdivision. In: ISCIS 02 Conference Proceedings, pp. 137–141.
Alexa, M., 2002. Split operators for triangular refinement. Computer Aided Geometric Design 19 (3), 169–172.
Catmull, E., Clark, J., 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided
Design 10, 350–355.
Claes, J., Beets, K., Van Reeth, F., 2002. A corner-cutting scheme for hexagonal subdivision surfaces. In: Shape Modeling
International 02 Conference Proceedings, pp. 13–20.
Deslauriers, G., Dubuc, S., 1989. Symmetric iterative interpolation processes. Constructive Approximation 5 (1), 49–68.
Dodgson, N.A., Sabin, M.A., Barthe, L., Hassan, M.F., 2002. Towards a ternary interpolating subdivision scheme for the
triangular mesh. Research Report UCAM-CL-TR-539. University of Cambridge Computer Laboratory.
Dodgson, N.A., 2004. An heuristic analysis of the classification of bivariate subdivision schemes. Research Report, University
of Cambridge Computer Laboratory.
Dodgson, N., Ivrissimtzis, I., Sabin, M., 2003. Characteristics of dual-√3 subdivision schemes. In: Cohen, A., Merrien, J.-L.,
Schumaker, L.L. (Eds.), Curve and Surface Fitting, Saint-Malo 2002. Nashboro Press, Brentville, TN.
Doo, D., Sabin, M., 1978. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10,
Dyn, N., Levin, D., Liu, D., 1992. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided
Design 24 (4), 211–216.
Dyn, N., Levin, D., Gregory, J.A., 1990. A butterfly subdivision scheme for surface interpolation with tension control. ACM
Transactions on Graphics 9 (2), 160–169.
Fowler, P.W., John, P.E., Sachs, H., 2000. (3, 6)-cages, hexagonal toroidal cages, and their spectra. In: Hansen, P., et al. (Eds.),
Discrete Mathematical Chemistry. In: Ser. Discrete Math. Theor. Comput. Sci., Vol. 51. DIMACS, pp. 139–174.
Ivrissimtzis, I.P., Dodgson, N.A., Hassan, M.F., Sabin, M.A., 2002a. On the geometry of recursive subdivision. Internat. J.
Shape Modeling 8 (1), 23–42.
Ivrissimtzis, I.P., Dodgson, N.A., Sabin, M.A., 2002b. Recursive subdivision and hypergeometric functions. In: Proceedings of
SMI2002: International Conference on Shape Modelling and Applications. IEEE Press, pp. 145–153.
Ivrissimtzis, I.P., Sabin, M.A., Dodgson, N.A., 2004. √5-subdivision. In: Advances in Multiresolution for Geometric
Modelling, submitted for publication.
Kobbelt, L., 1996. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics
Forum 15 (3), 409–420.
Kobbelt, L., 2000. √3-subdivision. In: Siggraph 2000, Conference Proceedings, pp. 103–112.
Labsik, U., Greiner, G., 2000. Interpolatory √3-subdivision. Computer Graphics Forum 19 (3), 131–138.
Loop, C.T., 1987. Smooth subdivision surfaces based on triangles. Master’s Thesis University of Utah, Department of
Loop, C., 2003. Smooth ternary subdivision of triangle meshes. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (Eds.), Curve
and Surface Fitting: Saint-Malo 2002. Nashboro Press, Brentville, TN.
Oswald, P., Schröder, P., 2003. Composite primal/dual √3-subdivision schemes. Computer Aided Geometric Design 20 (3),
Peters, J., Reif, U., 1997. The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics 16 (4),
Senechal, M., 1995. Quasicrystals and Geometry. Cambridge Univ. Press, Cambridge.
Sloan, I.H., Lyness, J.N., 1989. The representation of lattice quadrature rules as multiple sums. Math. Comput. 2 (185), 81–94.
Sloan, I., Joe, S., 1994. Lattice Methods for Multiple Integration. Oxford Science Publications. Clarendon Press, Oxford.
Stamminger, M., Drettakis, G., 2001. Interactive sampling and rendering for complex and procedural geometry. In: Proceedings
of the Eurographics Workshop on Rendering 01. Springer-Verlag, pp. 151–162.
Velho, L., 2001. Quasi 4-8 subdivision. Computer Aided Geometric Design 18 (4), 299–379.
Velho, L., 2003. Stellar subdivision grammars. In: Symposium on Geometry Processing, Eurographics/ACM SIGGRAPH
Symposium Proceedings. ACM, New York, pp. 188–199.
Velho, L., Zorin, D., 2001. 4-8 subdivision. Computer Aided Geometric Design 18 (5), 397–427.
Zhang, H.-X.,Wang, G.-J., 2002. Honeycomb subdivision. J. Software 13 (5), 1199–1208.
Zorin, D., Schröder, P., 2001. A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided
Geometric Design 18 (5), 429–454.