Abstract
Octrees are among the most widely used representations in geometric modeling systems, apart from Constructive Solid Geometry and Boundary Representations. An octree model is based on recursive cell decompositions of the space and does not depend greatly on the nature of the object but much more on the chosen maximum subdivision level. Unfortunately, an octree may require a large amount of memory when it uses a set of very small cubic nodes to approximate an object.
This paper is concerned with a novel generalization of the octree model that uses interval arithmetic and allows us to extend the tests for classifying points in space as inside, on or outside a CSG object to whole sections of the space at once. Tree nodes with additional information about relevant parts of the CSG object are introduced in order to reduce the depth of the required subdivision. The proposed extended octrees are compared with the common octree representation.
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References
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)
Brunet, P., Navazo, I.: Solid representation and operation using extended octrees. ACM Transactions on Graphics 9(2), 170–197 (1990)
Carlbom, I., Chakravarty, I., Vanderschel, D.: A Hierarchical Data Structure for Representing the Spatial Decomposition of 3D Objects. IEEE Computer Graphics and Applicalions 5(4), 24–31 (1985)
Duff, T.: Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry. Computer Graphics 26(2), 131–138 (1992)
Dupont, L., Lazard, S., Petitjean, S.: Towards the robust intersection of implicit quadrics. In: Proc. of Workshop on Uncertainty in Geometric Computations, Sheeld, UK (2001)
Dürst, M.J., Kunii, T.L.: Integrated polytrees: a generalized model for integrating spatial decomposition and boundary representation. Technical Report 88-002, Department of Information Science, Faculty of Science, University of Tokyo (1988)
Dyllong, E., Grimm, C.: Verified Adaptive Octree Representations of Constructive Solid Geometry Objects. In: Simulation und Visualisierung 2007, pp. 223–235. SCS Publishing House e. V, San Diego, Erlangen (2007)
Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: C++ Toolbox for Verified Computing. Basic Numerical Problems. Springer, Berlin (1995)
Hoffmann, C.M.: Geometric and Solid Modeling. Morgan Kaufmann, San Francisco (1989)
Lazarda, S., Penarandab, L.M., Petitjean, S.: Intersecting quadrics: an efficient and exact implementation. Computational Geometry 35(1-2), 74–99 (2006)
Levin, J.: Mathematical models for determining the intersections of quadric surfaces. Computer Graphics and Image Processing 11(1), 73–87 (1979)
Lennerz, C.: Distance Computation for Extended Quadratic Complexes. PhD Thesis, Faculty of Computer Science, University of Saarland (2005)
Lennerz, C., Schömer, E.: Efficient distance computation for quadratic curves and surfaces. In: Proc. of IEEE Geometric Modeling and Processing, Los Alamitos, USA, pp. 60–69 (2002)
Miller, J.R.: Geometric Approaches to Nonplanar Quadric Surface Intersection Curves. ACM Transactions on Graphics 6(4), 274–307 (1987)
Moore, R.: Interval Analysis. Prentice Hall, Englewood Cliffs (1966)
Ratschek, H., Rokne, J.: Geometric computations with interval and new robust methods: applications in computer graphics, GIS and computational geometry. Horwood Publishing, Chichester (2003)
Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley Publishing Company, Reading (1990)
Snyder, J.M.: Generative Modeling for Computer Graphics and CAD: Symbolic Shape Design Using Interval Analysis. Academic Press, San Diego (1992)
Wyvill, G., Kunii, T.L., Shirai, Y.: Space division for ray tracing in CSG. IEEE Computer Graphics and Applications 6(4), 28–34 (1986)
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Dyllong, E., Grimm, C. (2008). A Reliable Extended Octree Representation of CSG Objects with an Adaptive Subdivision Depth. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2007. Lecture Notes in Computer Science, vol 4967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68111-3_142
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DOI: https://doi.org/10.1007/978-3-540-68111-3_142
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