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A Study on Unifying Topology and Numerical Accuracy in Geometric Modeling: Surface to Surface Intersections  

Ko, Kwang-Hee (광주과학기술원)
Publication Information
Korean Journal of Computational Design and Engineering / v.12, no.5, 2007 , pp. 344-353 More about this Journal
Abstract
In this paper, we address the problem of robust geometric modeling with emphasis on surface to surface intersections. We consider the topology and the numerical accuracy of an intersection curve to find the best approximation to the exact one. First, we perform the topological configuration of intersection curves, from which we determine the starting and ending points of each monotonic intersection curve segment along with its topological structure. Next, we trace each monotonic intersection curve segment using a validated ODE solver, which provides the error bounds containing the topological structure of the intersection curve and enclosing the exact root without a numerical instance. Then, we choose one approximation curve and adjust it within the bounds by minimizing an objective function measuring the errors from the exact one. Using this process, we can obtain an approximate intersection curve which considers the topology and the numerical accuracy for robust geometric modeling.
Keywords
surface to surface intersection; geometric modeling; robustness; validated ODE solver; topology; accuracy;
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1 Farouki, R. T., 'Closing the Gap between CAD Model and Downstream Application', SIAM News, Vol. 32, No. 5, pp. 1-3, 1999
2 Sederberg, T. W., Implicit and Parametric Curves and Surfaces for Computer Aided Geometric Design, PhD Thesis, Purdue University, 1983
3 Katz, S. and Sederberg, T. W., 'Genus of the Intersection Curve of Two Rational Surface Patches', Computer Aided Geometric Design, Vol. 5, pp. 253-258, 1988   DOI   ScienceOn
4 Hoschek, J. and Lasser, D., Fundamentals of Computer Aided Geometric Design, A K Peters, Wellesley, MA, 1993
5 Bajaj, C. L. and Xu, G., 'NURBS Approximation of Surface/Surface Intersection Curves', Advances in Computational Mathematics, Vol. 2, pp. 1-21, 1994   DOI
6 Mukundan, H., Ko, K. H., Maekawa, T., Sakkalis, and Patrikalakis, N. M., 'Tracing Surface Intersections with a Validated ODE System Solver', Proceedings of the 9th EG/ACM Symposium on Solid Modeling and Applications, G. Elber, N. Patrikalakis and P. Brunet, editors. pp. 249-254. Genoa, Italy, Eurographics Press, 2004
7 Plegl, L. A. and Tiller, W., The Nurbs Book, Springer, New York, 1995
8 Zhu, C., Byrd, R. H., Lu, P. and Nocedal, J., LBFGS-B: FORTRAN Subroutines for Large Scale Bound Constrained Optimization. Tech. Report NAM-11, EECS Department, Northwestern University, 1994
9 Patrikalakis, N. M. and Maekawa, T., Shape Interrogation for Computer Aided Design and Manufacturing, Springer, New York, 2002
10 Song, X., Sederberg, T. W., Zheng, J., Farouki, R. T. and Hass, J., 'Linear Perturbation Methods for Topologically Consistent Representations of Free-form Surface Intersections', Computer Aided Geometric Design, Vol. 21, No. 3, pp. 303-319, 2004   DOI   ScienceOn
11 Park, H., 'An Error-bounded Approximate Method for Representing Planar Curves in B-splines', Computer Aided Geometric Design, Vol. 21, pp. 479-497, 2004   DOI   ScienceOn
12 Grandine, T. A. and Klein, F. W., 'A New Approach to the Surface Intersection Problem', Computer Aided Geometric Design, Vol. 14, pp. 111-134, 1997   DOI   ScienceOn
13 Nedialkov, N. S., Jackson, K. R. and Corliss, G. F., 'Validated Solutions of Initial Value Problems for Ordinary Differential Equations', Applied Mathematics and Computation, Vol. 105, No. 1, pp. 21-68, 1999   DOI   ScienceOn
14 Lohner, R. J., 'Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problem', In Computational Ordinary Differential Equations, Cash J., Gladwell I., (Eds.), Clarendon Press, Oxford, pp. 425-235, 1992
15 Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes in C, Cambridge University Press, Cambridge, 1988
16 Sakkalis, T., 'The Topological Configuration of a Real Algebraic Curve', Bulletin of the Australian Mathematical Society, Vol. 43, pp. 37-50, 1991   DOI
17 Patrikalakis, N. M., Maekawa, T., Ko, K. H. and Mukundan, H., 'Surface to Surface Intersections', Computer-Aided Design & Applications, Vol. 1, No. 1-4, pp. 449-458, 2004   DOI
18 Shen, G., Analysis of Boundary Representation Model Rectification, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA, 2000