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The detection of the number of disjoint components is a well-known procedure for surface objects. However, this problem has not been solved for solid models defined with scalar fields in the so-called implicit form. In this paper, we present a technique which allows for detection of the number of disjoint components with a predefined tolerance for an object defined with a single scalar function. The core of the technique is a reliable continuation of the spatial enumeration based on the interval methods. We also present several methods for separation of components using set-theoretic operations for further handling these components individually in a solid modelling system dealing with objects defined with scalar fields.
[1] B. Wyvill, A. Guy, E. Galin, Extending the CSG Tree. Warping, blending and Boolean operations in an implicit surface modeling system, Computer Graphics Forum 18 (2) (1999) 149-158.
[2] V. Adzhiev, R. Cartwright, E. Fausett, A. Ossipov, A. Pasko, V. Savchenko, HyperFun project: a framework for collaborative multidimensional F-Rep modelling, in: Proc Implicit Surfaces '99, Eurographics/ACM SIGGRAPH Workshop, J. Hughes and C. Schlick (Eds.), 1999, pp. 59-69.
[3] D. M. Mount, Intersection detection and separators for simple polygons, in: Proceedings of the eighth annual symposium on Computational geometry, SCG '92, ACM, New York, NY, USA, 1992, pp. 303-311.
[4] J. R. Rossignac, Solid and Physical Modeling, John Wiley & Sons, Inc., 2007.
[5] J. W. Boyse, Interference detection among solids and surfaces, Commun. ACM 22 (1) (1979) 3-9.
[6] R. B. Tilove, A null-object detection algorithm for constructive solid geometry, Commun. ACM 27 (7) (1984) 684-694.
[7] S. Bandi, D. Thalmann, An adaptive spatial subdivision of the object space for fast collision detection of animated rigid bodies, Computer Graphics Forum 14 (3) (1995) 259-270.
[8] J. C. Hart, Morse theory for implicit surface modeling, in: H.-C. Hege, K. Polthier (Eds.), Mathematical Visualization, Springer Berlin Heidelberg, 1998, pp. 257-268.
[9] B. T. Stander, J. C. Hart, Guaranteeing the topology of an implicit surface polygonization for interactive modeling, in: Proceedings of the 24th annual conference on Computer graphics and interactive techniques, SIGGRAPH '97, ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, 1997, pp. 279-286.
[10] E. Berberich, M. Kerber, M. Sagraloff, Exact geometrictopological analysis of algebraic surfaces, in: Proceedings of the twenty-fourth annual symposium on Computational geometry, SCG '08, ACM, New York, NY, USA, 2008, pp. 164-173.
[11] W. Harvey, O. Rbel, V. Pascucci, P.-T. Bremer, Y. Wang, Enhanced topology-sensitive clustering by Reeb graph shattering, in: R. Peikert, H. Hauser, H. Carr, R. Fuchs (Eds.), Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization, Springer Berlin Heidelberg, 2012, pp. 77-90.
[12] T. Nieda, A. Pasko, T. L. Kunii, Detection and classification of topological evolution for linear metamorphosis, The Visual Computer 22 (5) (2006) 346-356.
[14] J.-D. Boissonnat, J. Czyzowicz, O. Devillers, J. Urrutia, M. Yvinec, Computing largest circles separating two sets of segments, in: Proceedings of the 8th Canadian Conference on Computational Geometry, Carleton University Press, 1996, pp. 173-178.
[15] L. H. de Figueiredo, J. Stolfi, Self-Validated Numerical Methods and Applications, Brazilian Mathematics Colloquium monographs, IMPA/CNPq, Rio de Janeiro, Brazil, 1997.
[16] L. H. de Figueiredo, J. Stolfi, Affine arithmetic: Concepts and applications, Numerical Algorithms 37 (2004) 147-158.
[18] L. H. D. Figueiredo, J. Stolfi, Adaptive enumeration of implicit surfaces with affine arithmetic, Computer Graphics Forum 15 (1996) 287-296.
[19] J. Bloomenthal et al. (Ed.), Introduction to Implicit Surfaces, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1997.
[20] D. Kalra, A. H. Barr, Guaranteed ray intersections with implicit surfaces, SIGGRAPH Comput. Graph. 23 (3) (1989) 297-306.
[21] J. M. Snyder, Interval analysis for computer graphics, SIGGRAPH Comput. Graph. 26 (2) (1992) 121-130.
[22] A. Gomes, I. Voiculescu, J. Jorge, B. Wyvill, C. Galbraith, Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 1st Edition, Springer Publishing Company, Incorporated, 2009.
[23] D. Storti, C. Finley, M. Ganter, Interval extensions of signed distance functions: iSDF-reps and reliable membership classification, Journal of Computing and Information Science in Engineering 10 (2) (2010) 1-8.
[24] S. F. Frisken, R. N. Perry, A. P. Rockwood, T. R. Jones, Adaptively sampled distance fields: a general representation of shape for computer graphics, in: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, SIGGRAPH '00, ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, 2000, pp. 249-254.
[25] C. Ericson, Real-Time Collision Detection (The Morgan Kaufmann Series in Interactive 3D Technology), Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 2004.