Abstract
We describe a data structure for three-dimensional Nef complexes, algorithms for boolean operations on them, and our implementation of data structure and algorithms. Nef polyhedra were introduced by W. Nef in his seminal 1978 book on polyhedra. They are the closure of half-spaces under boolean operations and can represent non-manifold situations, open and closed boundaries, and mixed dimensional complexes. Our focus lies on the generality of the data structure, the completeness of the algorithms, and the exactness and efficiency of the implementation. In particular, all degeneracies are handled.
Work on this paper has been partially supported by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-2000-26473 (ECG – Effective Computational Geometry for Curves and Surfaces), and by the ESPRIT IV LTR Project No. 28155 (GALIA). We thank Sven Havemann and Peter Hoffmann for helpful discussions.
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
Agrawal, A., Requicha, A.G.: A paradigm for the robust design of algorithms for geometric modeling. Computer Graphics Forum 13(3), 33–44 (1994)
Banerjee, R., Rossignac, J.: Topologically exact evaluation of polyhedra defined in CSG with loose primitives. Computer Graphics Forum 15(4), 205–217 (1996)
Benouamer, M., Michelucci, D., Peroche, B.: Error-free boundary evaluation based on a lazy rational arithmetic: a detailed implementation. Computer-Aided Design 26(6) (1994)
Bieri, H.: Nef polyhedra: A brief introduct. Comp. Suppl. 10, 43–60 (1995)
Bieri, H.: Two basic operations for Nef polyhedra. In: CSG 1996: Set-theoretic Solid Modelling: Techniques and Applications, April 1996. Information Geometers, pp. 337–356(1996)
Bieri, H., Nef, W.: Elementary set operations with d-dimensional polyhedra. In: Noltemeier, H. (ed.) CG-WS 1988. LNCS, vol. 333, pp. 97–112. Springer, Heidelberg (1988)
Canny, J., Donald, B.R., Ressler, E.K.: A rational rotation method for robust geometric algorithms. In: Proc. ACM Sympos. Comput. Geom., pp. 251–260 (1992)
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introd. to Algorithms. MIT Press, Cambridge (1990)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Heidelberg (1997)
Dobrindt, K., Mehlhorn, K., Yvinec, M.: A complete and efficient algorithm for the intersection of a general and a convex polyhedron. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 314–324. Springer, Heidelberg (1993)
Fabri, A., Giezeman, G.-J., Kettner, L., Schirra, S., Schönherr, S.: On the design of CGAL a computational geometry algorithms library. Softw. – Pract. Exp. 30(11), 1167–1202 (2000)
Fortune, S.J.: Polyhedral modelling with multiprecision integer arithmetic. Computer-Aided Design 29, 123–133 (1997)
Gursoz, E.L., Choi, Y., Prinz, F.B.: Vertex-based representation of non-manifold boundaries. Geometric Modeling for Product Engineering 23(1), 107–130 (1990)
Halperin, D.: Robust geometric computing in motion. Int. J. of Robotics Research 21(3), 219–232 (2002)
Hemmer, M., Schömer, E., Wolpert, N.: Computing a 3-dimensional cell in an arrangement of quadrics: Exactly and actually. In: ACM Symp. on Comp. Geom., pp. 264–273 (2001)
Hoffmann, C.M.: Geometric and Solid Modeling – An Introd. Morgan Kaufmann, San Francisco (1989)
Karasick, M.: On the Representation and Manipulation of Rigid Solids. Ph.D. thesis, Dept. Comput. Sci., McGill Univ., Montreal, PQ (1989)
Kettner, L.: Using generic programming for designing a data structure for polyhedral surfaces. Comput. Geom. Theory Appl. 13, 65–90 (1999)
Keyser, J., Krishnan, S., Manocha, D.: Efficient and accurate B-rep generation of low degree sculptured solids using exact arithmetic. In: Proc. ACM Solid Modeling (1997)
Mäntylä, M.: An Introd. to Solid Modeling. Comp. Science Press, Rockville (1988)
Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999)
Middleditch, A.E.: The bug and beyond: A history of point-set regularization. In: CSG 1994 Set-theoretic Solid Modelling: Techn. and Appl., pp. 1–16. Inform. Geom. Ltd. (1994)
Nef, W.: Beiträge zur Theorie der Polyeder. Herbert Lang, Bern (1978)
Rossignac, J.R., O’Connor, M.A.: SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries. In: Wozny, M., Turner, J., Preiss, K. (eds.) Geometric Modeling for Product Engineering, North-Holland, Amsterdam (1989)
Rossignac, J.R., Requicha, A.G.: Solid modeling, http://citeseer.nj.nec.com/209266.html
Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading (1990)
Seel, M.: Implementation of planar Nef polyhedra. Research Report MPI-I-2001-1-2003, MPI für Informatik, Saarbrücken, Germany (August 2001)
Seel, M.: Planar Nef Polyhedra and Generic Higher-dimensional Geometry. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany, 5 (December 2001)
Seel, M., Mehlhorn, K.: Infimaximal frames: A technique for making lines look like segments. Comp. Geom. Theory and Appl. (2000) (to appear), http://www.mpi-sb.mpg.de/~mehlhorn/ftp/InfiFrames.ps
Stolfi, J.: Oriented Projective Geometry: A Framework for Geometric Computations. Academic Press, NewYork (1991)
Weiler, K.: The radial edge structure: A topological representation for non-manifold geometric boundary modeling. In: Wozny, M.J., McLaughlin, H.W., Encarnaçao, J.L. (eds.) Geom. Model. for CAD Appl., IFIP, May 12–16, pp. 3–36 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Granados, M., Hachenberger, P., Hert, S., Kettner, L., Mehlhorn, K., Seel, M. (2003). Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, and Implementation. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_59
Download citation
DOI: https://doi.org/10.1007/978-3-540-39658-1_59
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20064-2
Online ISBN: 978-3-540-39658-1
eBook Packages: Springer Book Archive