An algorithm for triangulating multiple 3D polygons
We present an algorithm for obtaining a triangulation of multiple, non‐planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a singl...
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| Veröffentlicht in: | Computer graphics forum Jg. 32; H. 5; S. 157 - 166 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2013
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We present an algorithm for obtaining a triangulation of multiple, non‐planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non‐manifold outputs for two and more input polygons without compromising optimality. For better performance on real‐world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality. |
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| AbstractList | We present an algorithm for obtaining a triangulation of multiple, non-planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non-manifold outputs for two and more input polygons without compromising optimality. For better performance on real-world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality. [PUBLICATION ABSTRACT] We present an algorithm for obtaining a triangulation of multiple, non-planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non-manifold outputs for two and more input polygons without compromising optimality. For better performance on real-world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality. |
| Author | Carr, Nathan Ju, Tao Zou, Ming |
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| References_xml | – reference: Ju T.: Fixing geometric errors on polygonal models: a survey. J. Comput. Sci. Technol. 24, 1 (Jan. 2009), 19-29. – reference: Bandyopadhyay D., Snoeyink J.: Almost-delaunay simplices: Robust neighbor relations for imprecise 3d points using cgal. Comput. Geom. Theory Appl. 38, 1-2 (Sept. 2007), 4-15. – reference: Gilbert P. D.: New results in planar triangulations. Tech. rep., Urbana , Illinois : Coordinated Science Laboratory, University of Illinois, 1979. – reference: Liu L., Bajaj C., Deasy J., Low D. A., Ju T.: Surface reconstruction from non-parallel curve networks. Comput. Graph. Forum 27, 2 (2008), 155-163. – reference: Attene M., Campen M., Kobbelt L.: Polygon mesh repairing: An application perspective. ACM Comput. Surv. 45, 2 (Mar. 2013), 15:1-15:33. – reference: Seidel R.: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Comput. Geom. 1 (1991), 51-64. – reference: Fuchs H., Kedem Z. M., Uselton S. P.: Optimal surface reconstruction from planar contours. Commun. ACM 20, 10 (Oct. 1977), 693-702. – reference: Bessmeltsev M., Wang C., Sheffer A., Singh K.: Design-driven quadrangulation of closed 3d curves. Transactions on Graphics (Proc. SIGGRAPH ASIA 2012) 31, 5 (2012). – reference: Barequet G., Sharir M.: Filling gaps in the boundary of a polyhedron. Comput. Aided Geom. Des. 12, 2 (Mar. 1995), 207-229. – reference: Chazelle B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6, 5 (Aug. 1991), 485-524. – reference: Barequet G., Goodrich M. T., Levi-Steiner A., Steiner D.: Straight-skeleton based contour interpolation. Graph. 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Des. – volume: 31 issue: 5 year: 2012 article-title: Design‐driven quadrangulation of closed 3d curves publication-title: Transactions on Graphics – volume: 6 start-page: 485 issue: 5 year: Aug. 1991 end-page: 524 article-title: Triangulating a simple polygon in linear time publication-title: Discrete Comput. Geom. – volume: 24 start-page: 19 issue: 1 year: Jan. 2009 end-page: 29 article-title: Fixing geometric errors on polygonal models: a survey publication-title: J. Comput. Sci. Technol. – volume: 65 start-page: 323 year: 2004 end-page: 350 article-title: Straight‐skeleton based contour interpolation publication-title: Graph. Models – year: 1979 – volume: 20 start-page: 693 issue: 10 year: Oct. 1977 end-page: 702 article-title: Optimal surface reconstruction from planar contours publication-title: Commun. ACM – volume: 27 start-page: 155 issue: 2 year: 2008 end-page: 163 article-title: Surface reconstruction from non‐parallel curve networks publication-title: Comput. Graph. Forum – volume: 79 start-page: 121 year: 1980 end-page: 123 – volume: 1 start-page: 51 year: 1991 end-page: 64 article-title: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons publication-title: Comput. Geom. – start-page: 121 volume-title: Combinatorics year: 1980 ident: e_1_2_8_15_2 – ident: e_1_2_8_21_2 – ident: e_1_2_8_10_2 doi: 10.1007/BF02574703 – ident: e_1_2_8_11_2 doi: 10.1145/359842.359846 – ident: e_1_2_8_12_2 doi: 10.1007/11602613_98 – ident: e_1_2_8_20_2 doi: 10.1016/0925-7721(91)90012-4 – ident: e_1_2_8_6_2 – start-page: 173 volume-title: Proceedings of the conference on Graphics interface '97 year: 1997 ident: e_1_2_8_19_2 – ident: e_1_2_8_17_2 – ident: e_1_2_8_16_2 doi: 10.1111/j.1467-8659.2008.01112.x – ident: e_1_2_8_3_2 doi: 10.1145/1449715.1449740 – ident: e_1_2_8_8_2 doi: 10.1016/j.comgeo.2006.11.003 – ident: e_1_2_8_22_2 – ident: e_1_2_8_7_2 doi: 10.1016/0167-8396(94)00011-G – volume: 31 issue: 5 year: 2012 ident: e_1_2_8_9_2 article-title: Design‐driven quadrangulation of closed 3d curves publication-title: Transactions on Graphics – ident: e_1_2_8_2_2 doi: 10.1145/2431211.2431214 – volume-title: New results in planar triangulations year: 1979 ident: e_1_2_8_13_2 – ident: e_1_2_8_4_2 doi: 10.1145/237218.237234 – ident: e_1_2_8_14_2 doi: 10.1007/s11390-009-9206-7 – volume: 65 start-page: 323 year: 2004 ident: e_1_2_8_5_2 article-title: Straight‐skeleton based contour interpolation publication-title: Graph. 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| SubjectTerms | 3-D graphics Algorithms Approximation Computer graphics Computer science Feeding Mathematical models Optimization Polygons Studies Three dimensional Triangles |
| Title | An algorithm for triangulating multiple 3D polygons |
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