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Issue No.02 - Feb. (2014 vol.20)
pp: 172-181
Hongwei Lin , Dept. of Math., Zhejiang Univ., Hangzhou, China
Yang Qin , Dept. of Math., Zhejiang Univ., Hangzhou, China
Hongwei Liao , Dept. of Math., Zhejiang Univ., Hangzhou, China
Yunyang Xiong , Dept. of Math., Zhejiang Univ., Hangzhou, China
ABSTRACT
Because the B-spline surface intersection is a fundamental operation in geometric design software, it is important to make the surface intersection operation robust and efficient. As is well known, affine arithmetic is robust for calculating the surface intersection because it is able to not only find every branch of the intersection, but also deal with some singular cases, such as surface tangency. However, the classical affine arithmetic is defined only for the globally supported polynomials, and its computation is very time consuming, thus hampering its usefulness in practical applications, especially in geometric design. In this paper, we extend affine arithmetic to calculate the range of recursively and locally defined B-spline basis functions, and we accelerate the affine arithmetic-based surface intersection algorithm by using a GPU. Moreover, we develop efficient methods to thin the strip-shaped intersection regions produced by the affine arithmetic-based intersection algorithm, calculate the intersection points, and further improve their accuracy. The many examples presented in this paper demonstrate the robustness and efficiency of this method.
INDEX TERMS
Graphics processing units, Splines (mathematics), Strips, Robustness, Accuracy, Three-dimensional displays, Acceleration,GPU acceleration, Graphics processing units, Splines (mathematics), Strips, Robustness, Accuracy, Three-dimensional displays, Acceleration, geometric design, Surface-surface intersection, affine arithmetic
CITATION
Hongwei Lin, Yang Qin, Hongwei Liao, Yunyang Xiong, "Affine Arithmetic-Based B-Spline Surface Intersection with GPU Acceleration", IEEE Transactions on Visualization & Computer Graphics, vol.20, no. 2, pp. 172-181, Feb. 2014, doi:10.1109/TVCG.2013.237
REFERENCES
 [1] C. Hu, T. Maekawa, N. Patrikalakis, and X. Ye, “Robust Interval Algorithm for Surface Intersections,” Computer-Aided Design, vol. 29, no. 9, pp. 617-627, 1997. [2] A. Krishnamurthy, R. Khardekar, S. McMains, K. Haller, and G. Elber, “Performing Efficient NURBS Modeling Operations on the GPU,” IEEE Trans. Visualization and Computer Graphics, vol. 15, no. 4, pp. 530-543, July/Aug. 2009. [3] M. Gleicher and M. Kass, “An Interval Refinement Technique for Surface Intersection,” Proc. Graphics Interface, pp. 242-249, 1992. [4] L. De Figueiredo, “Surface Intersection Using Affine Arithmetic,” Proc. Graphics Interface, pp. 168-175, 1996. [5] http://www.nvidia.com/objectcuda_home_new.html , 2013. [6] R. Moore, Interval Analysis. Prentice Hall, 1966. [7] K. Suffern and E. Fackerell, “Interval Methods in Computer Graphics,” Computers & Graphics, vol. 15, no. 3, pp. 331-340, 1991. [8] J. Snyder, “Interval Analysis for Computer Graphics,” ACM SIGGRAPH Computer Graphics, vol. 26, no. 2, pp. 121-130, 1992. [9] J. Comba and J. Stol, “Affine Arithmetic and Its Applications to Computer Graphics,” Proc. Brazilian Symp. Computer Graphics and Image Processing (SIBGRAPI '93), pp. 9-18, 1990. [10] H. Shou, R. Martin, I. Voiculescu, A. Bowyer, and G. Wang, “Affine Arithmetic in Matrix Form for Polynomial Evaluation and Algebraic Curve Drawing,” Progress in Natural Science, vol. 12, no. 1, pp. 77-81, 2002. [11] H. Shou, H. Lin, R. Martin, and G. Wang, “Modified Affine Arithmetic is More Accurate than Centered Interval Arithmetic or Affine Arithmetic,” Proc. IMA Int'l Conf. Math. of Surfaces, pp. 355-365, 2003. [12] H. Shou, H. Lin, R. Martin, and G. Wang, “Modified Affine Arithmetic in Tensor Form for Trivariate Polynomial Evaluation and Algebraic Surface Plotting,” J. Computational and Applied Math., vol. 195, nos. 1/2, pp. 155-171, 2006. [13] T. Sederberg and R. Farouki, “Approximation by Interval Bézier Curves,” IEEE Computer Graphics and Applications, vol. 12, no. 5, pp. 87-95, Sept. 1992. [14] C. Hu, T. Maekawa, E. Sherbrooke, and N. Patrikalakis, “Robust Interval Algorithm for Curve Intersections,” Computer-Aided Design, vol. 28, nos. 6/7, pp. 495-506, 1996. [15] H. Mukundan, K. Ko, T. Maekawa, T. Sakkalis, and N. Patrikalakis, “Tracing Surface Intersections with Validated ODE System Solver,” Proc. Ninth ACM Symp. Solid Modeling and Applications, pp. 249-254, 2004. [16] N. Patrikalakis, T. Maekawa, K. Ko, and H. Mukundan, “Surface to Surface Intersection,” Proc. Int'l CAD Conf. and Exhibition, vol. 4, 2004. [17] N. Aziz, R. Bata, and S. Bhat, “Surfaces: Bezier Surface/Surface Intersection,” IEEE Computer Graphics and Applications, vol. 10, no. 1, pp. 50-58, Jan. 1990. [18] D. Manocha and J. Canny, “A New Approach for Surface Intersection,” Int'l J. Computational Geometry and Applications, vol. 1, no. 4, pp. 491-516, 1991. [19] S. Krishnan and D. Manocha, “An Efficient Surface Intersection Algorithm Based on Lower-Dimensional Formulation,” ACM Trans. Graphics, vol. 16, no. 1, pp. 74-106, 1997. [20] D. Manocha and S. Krishnan, “Algebraic Pruning: A Fast Technique for Curve and Surface Intersection,” Computer Aided Geometric Design, vol. 14, no. 9, pp. 823-845, 1997. [21] G. Renner and V. Weiss, “Exact and Approximate Computation of B-Spline Curves on Surfaces,” Computer-Aided Design, vol. 36, no. 4, pp. 351-362, 2004. [22] D. Lasser, “Intersection of Parametric Surfaces in the Bernstein-Bézier Representation,” Computer-Aided Design, vol. 18, no. 4, pp. 186-192, 1986. [23] S. Briseid, T. Dokken, T. Hagen, and J. Nygaard, “Spline Surface Intersections Optimized for GPUs,” Proc. Sixth Int'l Conf. Computational Science (ICCS '06), pp. 204-211, 2006. [24] S. Briseid, T. Dokken, and T.R. Hagen, “Heterogeneous Spline Surface Intersections,” Proc. 26th Spring Conf. Computer Graphics (SCCG '10), pp. 141-148, http://doi.acm.org/10.11451925059.1925085 , 2010. [25] P. Sinha, E. Klassen, and K. Wang, “Exploiting Topological and Geometric Properties for Selective Subdivision,” Proc. First Ann. Symp. Computational Geometry, pp. 39-45, 1985. [26] D. Alcantara, A. Sharf, F. Abbasinejad, S. Sengupta, M. Mitzenmacher, J. Owens, and N. Amenta, “Real-Time Parallel Hashing on the GPU,” ACM Trans. Graphics, vol. 28, no. 5,article 154, 2009. [27] J. Hoschek and D. Lasser, Fundamentals of Computer-Aided Geometric Design. AK Peters, 1993. [28] T. Dokken, “Aspects of Intersection Algorithms and Approximation,” Doctor thesis, Univ. of Oslo, 1997. [29] L. De Figueiredo and J. Stolfi, “Affine Arithmetic: Concepts and Applications,” Numerical Algorithms, vol. 37, no. 1, pp. 147-158, 2004. [30] R. Martin, H. Shou, I. Voiculescu, A. Bowyer, and G. Wang, “Comparison of Interval Methods for Plotting Algebraic Curves,” Computer Aided Geometric Design, vol. 19, pp. 553-587, 2002. [31] N. Patrikalakis and T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing. Springer-Verlag, 2002. [32] H. Lin, W. Chen, and G. Wang, “Curve Reconstruction Based on an Interval B-Spline Curve,” The Visual Computer, vol. 21, no. 6, pp. 418-427, 2005. [33] http://www.sintef.nosisl, 2013.