Abstract
A method of flattening 3D triangulations for use in surface meshing is presented. The flattening method supports multiple boundary loops and directly produces planar locations for the vertices of the triangulation. The general nonlinear least-square fit condition for the triangle vertices includes conformal (angle preserving) and authalic (area preserving) conditions as special cases. The method of Langrange multipliers is used to eliminate rotational and translation degrees of freedom and enforce periodic boundary conditions. Using matrix partitioning, several alternative sets of constraints can be efficiently tested to find which produces the best domain. A surface boundary term is introduced to improve domain quality and break the symmetry of indeterminate multi-loop problems. The nonlinear problems are solved using a scaled conformal result as the initial input. The resulting 2D domains are used to generate 3D surface meshes. Results indicate that best mesh quality is achieved with domains generated using an intermediate altitude preserving condition. Apart from an admirable robustness and overall efficiency, the 2D developed domains are particularly suited for structured transfinite/mapped meshes which often reveal wiggly irregularities with most conventional developed domains. Flattening and meshing (both free and transfinite/mapped) results are presented for several 3D triangulations.
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
Floater, M.S., Hormann, K.: Surface parameterization: A tutorial and survey. In: Dodgson, M.S.E.N., Sabin, M. (eds.) Advances on Multiresolution in Geometric Modelling, Springer, Heidelberg (2004)
Floater, M.S.: Mean value coordinates. Computer Aided Geometric Design 14(3), 231–250 (1997)
Hormann, K., Greiner, G.: MIPS: An Efficient Global Parameterization Method. In: Laurent, J.-P., Sablonniere, P., Schumaker, L.L. (eds.) Proceedings, Curve and Surface Design, pp. 153–162. Vanderbilt University Press (2000)
Levy, B., Petitjean, S., Ray, N., Maillot, J.: Least Squares Conformal Maps for Automatic Texture Atlas Generation. In: Proceedings, ACM, SIGGRAPH (July 2002)
Desbrun, M., Meyer, M., Alliez, P.: Intrinsic Parameterization of Surface Meshes. Computer Graphics Forum (EUROGRAPHICS 2002) 21(3), 209–218 (2002)
Sheffer, A., de Sturler, E.: Parameterization of faceted surfaces for meshing using angle based flattening. Engineering with Computers 17, 326–337 (2001)
Sheffer, A., Levy, B., Mogilnitsky, M., Bogomyakov, A.: ABF++: Fast and Robust Angle Based Flattening. Proceedings, ACM Transactions on Graphics 24(2), 311–330 (2005)
Zayer, R., Levy, B. Seidel, H.-P.: Linear Angle Based Parameterization. In: Proceedings, Eurographics Symposium on Geometry Processing (2007)
Gu, X., Yau, S.-T.: Global Conformal Surface Parameterization. In: Proceedings, Eurographics Symposium on Geometry Processing, pp. 127–137 (2003)
Cabello, J.: Towards Quality Surface Meshing. In: Proceedings, 12th International Meshing Roundtable, pp. 201–213 (2003)
Schoof, A.J.G., Th, L.H., Van Beukering, M., Sluiter, M.L.C.: A general purpose two-dimensional mesh generator. Advances in Engineering Software 1(3), 131–136 (1979)
Mukherjee, N.: CSALF: A combined subdivision and advancing loop-front approach to generating controlled boundary structured free meshes. International Journal of CAD/CAM (submitted, 2007)
Perronnet, A.: Triangle, tetrahedron, pentahedron transfinite interpolations. Application to the generation of C0 or G1-continuous algebraic meshes. In: Proceedings. Internationa Conference of Numerical Grid Generation in Computational Field Simulations, Greenwich, England, July 6-9, 1998 vol. 7, pp. 467–476 (1998)
Mukherjee, N.: High Quality Bi-Linear Transfinite Meshing with Interior Point Constraints. In: Pebay, P.P. (ed.) Proceedings, 15th International Meshing Roundtable, pp. 309–323. Springer, Heidelberg (2006)
Lo, S.H.: A New Mesh Generation Scheme For Arbitrary Planar Domains. International Journal For Numerical Methods in Engineering (21), 1403–1426 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beatty, K., Mukherjee, N. (2008). Flattening 3D Triangulations for Quality Surface Mesh Generation. In: Garimella, R.V. (eds) Proceedings of the 17th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87921-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-87921-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87920-6
Online ISBN: 978-3-540-87921-3
eBook Packages: EngineeringEngineering (R0)