Abstract
Algorithms for many geometric queries rely on representations that are comprised of combinatorial (logical, incidence) information, usually in a form of a graph or a cell complex, and geometric data that represents embeddings of the cells in the Euclidean space E d. Whenever geometric embeddings are imprecise, their incidence relationships may become inconsistent with the associated combinatorial model. Tolerant algorithms strive to compute on such representations despite the inconsistencies, but the meaning and correctness of such computations have been a subject of some controversy.
This paper argues that a tolerant algorithm usually assumes that the approximate geometric representation corresponds to a subset of E d that is homotopy equivalent to the intended exact set. We show that the Nerve Theorem provides systematic means for identifying sufficient conditions for the required homotopy equivalence, and explain how these conditions are used in the context of geometric and solid modeling.
Based on the talk at the Dagstuhl Seminar on Reliable Implementation of Real Number Algorithms: Theory and Practice, January 8-13, 2006.
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References
Requicha, A.A.G.: Representations for rigid solids: Theory, methods and systems. ACM Computing Surveys 12(4), 437–464 (1980)
Shapiro, V.: Solid modeling. In: Farin, G., Hoschek, J., Kim, M.S. (eds.) Handbook of Computer Aided Geometric Design, pp. 473–518. Elsevier Science Publishers, Amsterdam (2002)
Shapiro, V.: Maintenance of geometric representations through space decompositions. International Journal of Computational Geometry and Applications 7(4), 383–418 (1997)
Qi, J., Shapiro, V., Stewart, N.F.: Single-set and class-of-sets semantics for geometric models. Technical Report 2005-1, Spatial Automation Laborotary, University of Wisconsin - Madison (2005)
Tilove, R.B.: Set membership classification: A unified approach to geometric intersection problems. IEEE Transactions on Computer C-29(10) (1980)
Qi, J., Shapiro, V.: Epsilon-solidity in geometric data translation. Technical report, SAL 2004-2, Spatial Automation Laboratory, University of Wisconsin-Madison (2004)
Kettner, L., Pion, K.M., Schirra, S., Yap, S.: Classroom examples of robustness problems in geometric computations. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 702–713. Springer, Heidelberg (2004)
Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Heidelberg (2002)
Sabin, M.: Subdivision surfaces. In: Farin, G., Hoschek, J., Kim, M.S. (eds.) Handbook of Computer Aided Geometric Design, pp. 309–341. Elsevier Science Publishers, Amsterdam (2002)
Peters, J., Wu, X.: SLEVEs for planar spline curves. Computer Aided Geometric Design 21, 615–635 (2004)
Segal, M., Sequin, C.: Consistent calculations for solids modeling. In: SCG 1985: Proceedings of the first annual symposium on Computational geometry, pp. 29–38. ACM Press, New York (1985)
Segal, M.: Using tolerances to guarantee valid polyhedral modeling results. In: Computer Graphics (Proceedings of ACM SIGGRAPH 1990), pp. 105–114 (1990)
Jackson, D.J.: Boundary representation modelling with local tolerancing. In: Proceedings of the 3rd ACM Symposium on Solid Modeling and Applications, Salt Lake City, Utah, pp. 247–253 (1995)
Fang, S., Bruderlin, B., Zhu, X.: Robustness in solid modeling: A tolerance-based intuitionistic approach. Computer-Aided Design 25(9), 567–576 (1993)
Guibas, L., Salesin, D., Stolfi, J.: Epsilon geometry: Building robust algorithms from imprecise computations. In: Proceedings of the fifth ACM Symposium on Computational Geometry, Saarbruchen, West, Germany, pp. 208–217 (1989)
Shen, G., Sakkalis, T., Patrikalakis, N.M.: Analysis of boundary representation model rectification. In: Proceedings of the 6th ACM Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, pp. 149–158 (2001)
Chazal, F., Cohen-Steiner, D.: A condition for isotopic approximation. In: Proceedings of the 2004 ACM Symposium on Solid Modeling and Applications, Genova, Italy (2004)
Andersson, L.E., Stewart, N.F., Zidani, M.: Error analysis for operations in solid modeling (2004), www.iro.umontreal.ca/~stewart
Hoffmann, C.M., Stewart, N.F.: Accuracy and semantics in shape-interrogation applications. Graphical Models (to appear, 2005)
Song, X., Sederberg, T.W., Zheng, J., Farouki, R.T., Hass, J.: Linear perturbation methods for topologically consistent representations of free-form surface intersections. Computer Aided Geometric Design 21(3), 303–319 (2004)
O’Connor, M.A., Rossignac, J.R.: SGC: A dimension independent model for pointsets with internal structures and incomplete boundaries. In: IFIP/NSF Workshop on Geometric Modeling, Rensselaerville, NY, 1988, North-Holland, Amsterdam (1990)
Grandine, T.A., Frederick, W., Klein, I.: A new approach to the surface intersection problem. Comput. Aided Geom. Des. 14(2), 111–134 (1997)
Sakkalis, T., Peters, T.J., Bisceglio, J.: Isotopic approximations and interval solids. Computer-Aided Design 36, 1089–1100 (2004)
Yap, C.: Robust geometric computation. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton (1997)
Sugihara, K., Iri, M., Inagaki, H., Imai, T.: Topology-oriented implementation - an approach to robust geometric algorithms. Algorithmica 27(1), 5–20 (2000)
Bjorner, A.: Topological methods. In: Graham, R., Grotschel, M., Lovacz, L. (eds.) Handbook of Combinatorics, pp. 1819–1872. Elsevier Science B.V, Amsterdam (1995)
Alexandrov, P.: Gestalt u. lage abgeschlossener menge. The Annals of Mathematics 30, 101–187 (1928)
Hocking, J.G., Young, G.S.: Topology. Dover Publications, New York (1961)
Dey, T., Edelsbrunner, H., Guha, S.: Computational topology. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry (Contemporary mathematics 223), pp. 109–143. American Mathematical Society (1999)
Maunder, C.R.F.: Algebraic Topology. Dover Publications, New York (1996)
Mehlhorn, K., Yap, C.: Robust Geometric Computation (tentative). Book draft, under preparation (2004), http://www.cs.nyu.edu/~yap/book/egc/
Egenhofer, M., Clementini, E., di Felice, P.: Evaluating inconsistencies among multiple representations. In: Sixth International Symposium on Spatial Data Handling, Edinburgh, Scotland, pp. 901–920 (1994)
Kang, H.K., Kim, T.W., Li, K.J.: Topological consistency for collapse operation in multi-scale databases. In: Wang, S., Tanaka, K., Zhou, S., Ling, T.-W., Guan, J., Yang, D.-q., Grandi, F., Mangina, E.E., Song, I.-Y., Mayr, H.C. (eds.) ER Workshops 2004. LNCS, vol. 3289, pp. 91–102. Springer, Heidelberg (2004)
Armstrong, C.G.: Integrating analysis and design - thoughts for the future. In: State of the Art in CAD/FE Integration - NAFEMS Awareness Seminar, Chester, UK (2002), http://sog1.me.qub.ac.uk/Resources/publications/publications.php
Qi, J., Shapiro, V.: Epsilon-topological formulation of tolerant solid modeling. Computer-Aided Design 38(4), 367–377 (2006)
Sakkalis, T., Shen, G., Patrikalakis, N.M.: Topological and geometric properties of interval solid models. Graphical Models 63(3), 163–175 (2001)
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Shapiro, V. (2008). Homotopy Conditions for Tolerant Geometric Queries. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds) Reliable Implementation of Real Number Algorithms: Theory and Practice. Lecture Notes in Computer Science, vol 5045. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85521-7_10
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DOI: https://doi.org/10.1007/978-3-540-85521-7_10
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