Abstract
We introduce a general framework for sweeping a set of curves embedded on a two-dimensional parametric surface. We can handle planes, cylinders, spheres, tori, and surfaces homeomorphic to them. A major goal of our work is to maximize code reuse by generalizing the prevalent sweep-line paradigm and its implementation so that it can be employed on a large class of surfaces and curves embedded on them. We have realized our approach as a prototypical Cgal package. We present experimental results for two concrete adaptations of the framework: (i) arrangements of arcs of great circles embedded on a sphere, and (ii) arrangements of intersection curves between quadric surfaces embedded on a quadric.
This work has been supported in part by the IST Programme of the EU as Shared-cost RTD (FET Open) Project under Contract No IST-006413 (ACS - Algorithms for Complex Shapes), by the Israel Science Foundation (grant no. 236/06), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.
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
Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. Elsevier, Amsterdam (2000)
Andrade, M.V.A., Stolfi, J.: Exact algorithms for circles on the sphere. Internat. J. Comput. Geom. Appl. 11(3), 267–290 (2001)
Bentley, J.L., Ottmann, T.: Algorithms for reporting and counting geometric intersections. IEEE Trans. Computers 28(9), 643–647 (1979)
Berberich, E., Eigenwillig, A., Hemmer, M., Hert, S., Kettner, L., Mehlhorn, K., Reichel, J., Schmitt, S., Schömer, E., Wolpert, N.: Exacus: Efficient and exact algorithms for curves and surfaces. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 155–166. Springer, Heidelberg (2005)
Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R.: A general framework for processing a set of curves defined on a continuous 2D parametric surface (2007), http://www.cs.tau.ac.il/cgal/Projects/arr_on_surf.php
Berberich, E., Hemmer, M., Kettner, L., Schömer, E., Wolpert, N.: An exact, complete and efficient implementation for computing planar maps of quadric intersection curves. In: Proc. 21st SCG, pp. 99–106 (2005)
Cazals, F., Loriot, S.: Computing the exact arrangement of circles on a sphere, with applications in structural biology. Technical Report 6049, Inria Sophia-Antipolis (2006)
Fogel, E., Halperin, D.: Exact and efficient construction of Minkowski sums of convex polyhedra with applications. In: Proc. 8th ALENEX (2006)
Fogel, E., Halperin, D., Kettner, L., Teillaud, M., Wein, R., Wolpert, N.: Arrangements. In: Boissonnat, J.-D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces, vol. ch. 1, pp. 1–66. Springer, Heidelberg (2006)
Halperin, D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., ch. 24, pp. 529–562. Chapman & Hall/CRC (2004)
Halperin, D., Shelton, C.R.: A perturbation scheme for spherical arrangements with application to molecular modeling. Comput. Geom. Theory Appl. 10, 273–287 (1998)
Mehlhorn, K., Seel, M.: Infimaximal frames: A technique for making lines look like segments. J. Comput. Geom. Appl. 13(3), 241–255 (2003)
Meyerovitch, M.: Robust, generic and efficient construction of envelopes of surfaces in three-dimensional space. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 792–803. Springer, Heidelberg (2006)
Snoeyink, J., Hershberger, J.: Sweeping arrangements of curves. In: Proc. 5th SCG, pp. 354–363 (1989)
Wein, R., Fogel, E., Zukerman, B., Halperin, D.: 2D arrangements. In: Cgal-3.3 User and Reference Manual (2007), http://www.cgal.org/Manual/3.3/doc_html/cgal_manual/Arrangement_2/Chapter_main.html
Wein, R., Fogel, E., Zukerman, B., Halperin, D.: Advanced programming techniques applied to Cgal’s arrangement package. Comput. Geom. Theory Appl. 38(1–2), 37–63 (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berberich, E., Fogel, E., Halperin, D., Mehlhorn, K., Wein, R. (2007). Sweeping and Maintaining Two-Dimensional Arrangements on Surfaces: A First Step. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_57
Download citation
DOI: https://doi.org/10.1007/978-3-540-75520-3_57
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75519-7
Online ISBN: 978-3-540-75520-3
eBook Packages: Computer ScienceComputer Science (R0)