Abstract
Various Partial Differential Equations (PDEs) have been used in computer graphics for approximating surfaces of geometric shapes by finding solutions to PDEs, subject to suitable boundary conditions. The PDE boundary conditions are defined as 3D curves on surfaces of the shapes. We propose how to automatically derive these curves from the surface of the original polygon mesh. Analytic solutions to the PDEs used throughout this work are fully determined by finding a set of coefficients associated with parametric functions according to the particular set of boundary conditions. When large polygon meshes are used, the PDE coefficients require an order of magnitude smaller space compared to the original polygon data and can be interactively rendered with different levels of detail. It allows for an efficient exchange of the PDE shapes in 3D Cyberworlds and their web visualization. In this paper we analyze and formulate the requirements for extracting suitable boundary conditions, describe the algorithm for the automatic deriving of the boundary curves, and present its implementation as a part of the function-based extension of VRML and X3D.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Sheng, Y., Sourin, A., Gonzalez Castro, G., Ugail, H.: A PDE Method for Patchwise Approximation of Large Polygon Meshes. The Visual Computer 26(6-8), 975–984 (2010)
Ugail, H., Sourin, A.: Partial Differential Equations for Function based Geometry Modelling within Visual Cyberworlds. In: 2008 Int Conf. on Cyberworlds, pp. 224–231. IEEE CS, Los Alamitos (2008)
Ugail, H., González Castro, G., Sourin, A., Sourina, O.: Towards a Definition of Virtual Objects with Partial Differential Equations. In: 2009 Int. Conf. on Cyberworlds, Bradford, September 7-11, pp. 138–145 (2009)
Bloor, M., Wilson, M.: Generating blend surface using partial differential equations. Computer Aided Design 21(3), 165–171 (1989)
Schroeder, W., Zarge, J., Lorensen, W.: Decimation of triangle meshes. Computer Graphics 26(2), 65–70 (1992)
Hoppe, H., DeRose, T., Duchamp, T., et al.: Mesh optimization. In: SIGGRAPH 1993, pp. 19–26 (1993)
Rossignac, J., Borrel, P.: Multi-resolution 3D approximations for rendering complex scenes. In: Falcidieno, B., Kunii, T. (eds.) Modeling in Computer Graphics: Methods Application, pp. 455–465 (1993)
Turk, G.: Re-tiling polygonal surfaces. In: SIGGRAPH 1992, pp. 55–64 (1992)
Lounsbery, M.: Multiresolution analysis for surfaces of arbitrary topological type [PhD dissertation], Department of Computer Science and Engineering, University of Washington (1994)
Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: SIGGRAPH 1997, pp. 209–216 (1997)
Melia, S.: A simple, fast, effective polygon reduction algorithm. Game Developer 10, 44–49 (1998)
Levy, B.: Parameterization and deformation analysis on a manifold. Technical report, Alice (2007), http://alice.loria.fr/publications
Floater, M.: Mean value coordinates. Computer Aided Geometric Design 20(1), 19–27 (2003)
Tutte, W.: How to draw a graph. Proc of the London Mathematical Society, 743–768 (1963)
Chen, G.L., Pang, M.Y., Wang, J.D.: Calculating shortest path on edge-based data structure of graph. In: 2nd International Workshop on Digital Media and its Application in Museum and Heritage, pp. 416–421 (2007)
Chen, J., Han, Y.: Shortest paths on a polyhedron, Part I: Computing shortest paths. International Journal of Computer Geometry Application 6, 127–144 (1996)
Kaneva, B., O’Rourke, J.: An implementation of chen & han’s shortest paths algorithm. In: 21st Canadian Conference on Computer Geometry, pp. 139–146 (2000)
Kapoor, S.: Efficient computation of geodesic shortest paths. In: 31st ACM Symposium on Theory of Computing, pp. 770–779 (1999)
Lanthier, M., Maheshwari, A., Sack, J.R.: Approximating weighted shortest paths on polyhedral surfaces. In: 13th Annual Symposium on Computational Geometry, pp. 274–283 (1997)
Martínez, D., Velho, L., Carvalho, P.: Geodesic Paths on Triangular Meshes. In: SIBGRAPI/SIACG, pp. 210–217 (2004)
Mitchell, J.: Geometric shortest paths and network optimization. In: Sack, J., Urrutia, J. (eds.) Handbook of Computational Geometry, Elsevier Science, pp. 633–702 (2000)
Surazhsky, V., Surazhsky, T., Kirsanov, D., et al.: Fast exact and approximate geodesics on meshes. ACM Transactions on Graphics 24(3), 553–560 (2005)
Sourin, A., Wei, L.: Visual Immersive Haptic Mathematics. Virtual Reality 13(4), 221–234 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pang, MY., Sheng, Y., Sourin, A., González Castro, G., Ugail, H. (2011). Reconstructing Multiresolution Mesh for Web Visualization Based on PDE Resampling. In: Gavrilova, M.L., Tan, C.J.K., Sourin, A., Sourina, O. (eds) Transactions on Computational Science XII. Lecture Notes in Computer Science, vol 6670. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22336-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-22336-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22335-8
Online ISBN: 978-3-642-22336-5
eBook Packages: Computer ScienceComputer Science (R0)