Abstract
Given a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides view-dependent refinement and facilitates the computation of parameterization. For a triangulated closed surface of n vertices and genus g, we prove that there is a constant c > 0 such that if n > c.g, a greedy strategy can identify Θ(n) topology-preserving edge contractions that do not interfere with each other. Further, each of them affects only a constant number of triangles. Repeatedly identifying and contracting such edges produces a topology-preserving hierarchy of O(n + g 2) size and O(logn + g) depth. When no contractible edge exists, the triangulation is irreducible. Nakamoto and Ota showed that any irreducible triangulation of an orientable 2-manifold has at most maxû342g - 72,4ý vertices. Using our proof techniques we obtain a new bound of max|240g, 4.
Research of the first and third authors was partially supported by RGC CERG HKUST 6088/99E and HKUST 6190/02E.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D.W. Barnette and A.L. Edelson, All 2-manifolds have finitely many minimal triangulations. Israel J. Math, 67 (1989), 123–128.
J. Cohen, A. Varshney, D. Manocha, G. Turk, H. Weber, P. Agarwal, F. Brooks, and W. Wright. Simplification Envelopes. Proceedings of SIGGRAPH 96, 1996, 119–128.
M. de Berg and K.T.G. Dobrindt. On levels of detail in terrains. Proc. ACM Symposium on Computational Geometry, 26–27, 1995.
L. De Floriani, P. Magillo and E. Puppo. Building and traversing a surface at variable resolution. Proc. IEEE Visualization’ 97, 103–110, 1997.
T.K. Dey, H. Edelsbrunner, S. Guha and D.V. Nekhayev, Topology preserving edge contraction, Publ. Inst. Math. (Beograd) (N.S.), 66 (1999), 23–45.
A. Dold. Lectures on Algebraic Topology, Springer-Verlag, 1972.
C.A. Duncan, M.T. Goodrich, and S.G. Kobourov, Planarity-preserving clustering and embedding for large planar graphs, Proc. Graph Drawing’ 99, 186–196.
M. Garland and P.S. Heckbert. Surface simplification using quadric error metrics. Proc. SIGGRAPH’ 97, 209–216.
P.J. Gilbin. Graphs, Surfaces and Homology. Chapman and Hall, 1981.
H. Hoppe, Progressive meshes. Proc. SIGGRAPH’ 96, 99–108.
H. Hoppe, View-dependent refinement of progressive meshes. Proc. SIGGRAPH’ 97, 189–198.
D.G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12 (1983), 28–35.
A. Lee, W. Sweldens, P. Schröder, L. Cowsar and D. Dobkin, MAPS: multiresolution adaptive parameterization of surfaces. Proc. SIGGRAPH’ 98, 95–104.
A. Nakamoto and K. Ota, Note on irreducible triangulations of surfaces. Journal of Graph Theory, 10 (1995), 227–233.
J.C. Xia, J. El-Sana and A. Varshney. Adaptive real-time level-of-detail-based rendering for polygonal models. IEEE Transactions on Visualization and Computer Graphics, 3 (1997), 171–181.
J.C. Xia and A. Varshney, Dynamic view-dependent simplification for polygonal models. Proc. Visualization’ 96, 327–334.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cheng, SW., Dey, T.K., Poon, SH. (2002). Hierarchy of Surface Models and Irreducible Triangulation. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_26
Download citation
DOI: https://doi.org/10.1007/3-540-36136-7_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00142-3
Online ISBN: 978-3-540-36136-7
eBook Packages: Springer Book Archive