Abstract
The number type leda_real provides exact computation for a subset of real algebraic numbers: Every integer is a leda_real, and leda_reals are closed under the basic arithmetic operations +, -, *, / and k-th root operations. leda_reals guarantee correct results in all comparison operations. The number type is available as part of the LEDA C++ software library of efficient data types and algorithms (LEDA, Mehlhorn and Nüher 2000). leda_reals provide user-friendly exact computation. All the internals are hidden to the user. A user can use leda_reals just like any buHt-in number type. The number type is successfully used to solve precision and robustness problems in geometric computing (Burnikel et al. 2000, Seel). It is particularly advantageous when used in combination with the computational geometry algorithms library CGAL.
This is a preview of subscription content, log in via an institution to check access.
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
Brönniman, H., Kettner, L., Schirra, S., Veltkamp, R. (1998): Applications of the generic programming paradigm in the design of CGAL. Research Report MPI-I-98-1-030, Max-Planck-Insitut für Informatik
Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S. (1999): Companion page to ‚Efficient exact geometric computation made easy‘. http:/www.mpi-sb.mpg.de/~stschirr/exact/made_easy
Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S. (2000): A strong and easily computable separation bound for arithmetic expressions involving radicals. Algorithmica 27:87–99
Burnikel, C., Mehlhorn, K., Schirra, S. (1996): The LEDA class real number. Research Report MPI-I-96-1-001, Max-Planck-Institut für Informatik. A more recent documentation ofthe implementation is available at http://www.mpi-sb.mpg.de/~burnikel/reports/real.ps.gz.
CGAL:http://www.cgal.org
Fabri, A., Giezeman, G.-J., Kettner, L., Schirra, S., Schönherr, S. (1998): On the design of CGAL, the computational geometry algorithms library. Research Report MPI-I-98-1-007, Max-Planck-Institut für Informatik
Fortune, S. (1993): Progress in computational geometry. In: Martin R., (ed.): Directions in Geometric Computing, pp. 81–128. Information Geometers Ltd.
Hanniel, I., Halperin, D. (2000): Two-dimensional arrangements in CGAL and adaptive point location for parametrie curves. Manuscript. Tel-Aviv University, Israel
Hoffmann, C. (1989): The problem of accuracy and robustness in geometric computation. IEEE Computer 3:31–41
Karamcheti, V., Li, C., Pechtanski, I., Yap, C.K. (1999): A core library for robust numeric and geometric computation. In: Proc. 15th Annu. ACM Sympos. Comput. Geom., pp. 351–359
Keyser, J., Culver, T., Manocha, D., Krishnan, S. (1999): MAPC: A library for efficient and exact manipulation of algebraic points and curves. In: Proc. 15th Annu. ACM Sympos. Comput. Geom., pp. 360–369. See also http://www.cs.unc.edu/~geom/MAPC
LEDA:http://www.mpi-sb.mpg.de/LEDA
Megiddo, N. (1983): Applying parallel eomputation algorithms in the design of serial algorithms. Journal of the ACM 30:852–865
Mehlhorn, K, Nüher, S. (1994): The implementation of geometric algorithms. In: Proceedings of the 13th IFIP World Computer Congress, vol. 1, pp. 223–231. Elsevier Science B.V. North-Holland, Amsterdam
Mehlhorn, K, Nüher, S. (2000):LEDA–A Platform for Combinatorial and Geometric Computing. Cambridge University Press
Mnev, M.E. (1989): The universality theorems on the classification problem of configuration varieties and eonvex polytopes varieties. In: Topology and Geometry-Rohlin Seminar, Springer, pp. 527–544 (Lecture Notes Math., vol. 1346)
Overmars, M. (1996): Designing the computational geometry algorithms library CGAL. In Lin, M.C., Manocha, D., (eds.): Applied Computational Geometry: Towards Geometric Engineering (WACG96), Springer, pp. 53–58 (Lecture Notes in Computer Science, vol. 1148)
Preparata, F.P., Shamos, M.I. (1985): Computational Geometry: An Introduction. Springer
Rege, A. (1996): APU User Manual–Version 2.0. http://www.cs.berkeley.edu/~rege/apu/apu.html
Salowe, J. (1997): Parametric search. In: Goodman, J.E., Rourke, J.O. (eds.): Handbook of Discrete and Computational Geometry, pp. 683–698, CRC Press
Schirra, S. (2000): Robustness and Precision Issues in Geometric Computation. In: Sack, J.-R., Urrutia, J. (eds.): Handbook of Computational Geometry, Elsevier, pp. 597–632
Schirra, S. (1999): A case study on the cost of geometric eomputing. In: McGeoeh C.G, Goodrich M. (eds.): Algorithm Engineering and Experimentation (ALENEX’99), Springer pp. 156–176 (Lecture Notes in Computer Science, vol. 1619)
Seel, M.: Abstract Voronoi Diagrams. http://www.mpi-sb.mpg.de/LEDA/friends/avd.html
Schwerdt, J., Smid, M., Schirra, S. (1997): Computing the Minimum Diameter for Moving Points: An Exact Implementation using Parametric Search. In: Proc. of the 13th ACM Symp. on Computational Geometry, ACM Press, pp. 466–468.
Yap, C.K (1997a): Towards exact geometric computation. Comput. Geom. Theory Appl., 7:3–23
Yap, C.K (1997b): Robust geometric computation. In: Goodman, J.E., Rourke, J.O. (eds.): Handbook of Discrete and Computational Geometry, pp. 653–668, CRC Press
Yap, C.K, Dubé, T. (1995): The exact eomputation paradigm. In: Du, D., Hwang, F. (eds.), Computing in Euclidean Geometry, 2nd edition, World Scientific Press, pp. 452–492
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Wien
About this paper
Cite this paper
Mehlhorn, K., Schirra, S. (2001). Exact Computation with leda_real — Theory and Geometrie Applications. In: Alefeld, G., Rohn, J., Rump, S., Yamamoto, T. (eds) Symbolic Algebraic Methods and Verification Methods. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6280-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6280-4_16
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83593-7
Online ISBN: 978-3-7091-6280-4
eBook Packages: Springer Book Archive