Abstract
We compare approaches to the location of the algebraic curve f(x,y) = 0 in a rectangular region of the plane, based on recursive use of conservative estimates of the range of the function over a rectangle. Previous work showed that performing interval arithmetic in the Bernstein basis is more accurate than using the power basis, and that affine arithmetic in the power basis is better than using interval arithmetic in the Bernstein basis. This paper shows that using affine arithmetic with the Bernstein basis gives no advantage over affine arithmetic with the power basis. It also considers the Bernstein coefficient method based on the convex hull property, which has similar performance to affine arithmetic.
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References
Berchtold, J., The Bernstein Form in Set-Theoretic Geometric Modelling, PhD Thesis, University of Bath, 2000.
Berchtold, J., Bowyer, A., Robust Arithmetic for Multivariate Bernstein-Form Polynomials, Computer Aided Design, 2000, 32: 681–689.
Bowyer, A., Berchtold, J., Eisenthal, D., Voiculescu, I., and Wise, K., Interval Methods in Geometric Modelling, Geometric Modeling and Processing 2000, Eds. Martin, R., Wang, W., IEEE Computer Society Press, 2000, 321–327.
Chandler, R. E., A Tracking Algorithm for Implicitly Defined Curves, IEEE Computer Graphics & Applications, 1988, 8(2): 83–89.
Comba, J. L. D., Stolfi, J., Affine Arithmetic and its Applications to Computer Graphics, Anais do VII SIBGRAPI (Brazilian Symposium on Computer Graphics and Image Processing), Recife, Brazil, 1993, 9–18.
Duff, T, Interval Arithmetic and Recursive Subdivision for Implicit Functions and Constructive Solid Geometry, Computer Graphics (SIG-GRAPH’92 Proceedings), 1992, 26(2): 131–138.
R. T. Farouki, V. T. Rajan, On the Numerical Condition of Polynomials in Bernstein Form, Computer Aided Geometric Design, 191–216, 1987.
de Figueiredo, L. H., Surface Intersection Using Affine Arithmetic, Proceedings of Graphics Interface’96, Eds. MacKenzie S., Stewart J., Morgan Kaufmann, 1996, 168–175.
de Figueiredo, L. H., Stolfi, J., Adaptive Enumeration of Implicit Surfaces with Affine Arithmetic, Computer Graphics Forum, 1996, 15(5): 287–296.
Heidrich, W., Slusallek, P., Seidel, H. P., Sampling of Procedural Shaders using Affine Arithmetic, ACM Transaction on Graphics, 1998, 17(3): 158–176.
Moore, R. E., Methods and Applications of Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia, 1979.
Mudur, S. P., Koparkar, P A., Interval Methods for Processing Geometric Objects, IEEE Computer Graphics & Applications, 1984, 4(2): 7–17.
Ratschek, H., Rokne, J., Computer Methods for the Range of Functions, Ellis Horwood Ltd., 1984.
Shou H., Martin R., Voiculescu I., Bowyer A., Wang G., Affine Arithmetic in Matrix Form for Algebraic Curve Drawing, Progress in Natural Science, 12 (1)77–81, 2002.
Snyder, J. M., Interval Analysis for Computer Graphics, Computer Graphics (SIGGRAPH’92 Proceedings), 1992, 26(2): 121–130.
Suffern, K. G., Quadtree Algorithms for Contouring Functions of Two Variables, The Computer Journal, 1990, 33: 402–407.
Suffern, K. G., Fackerell, E. D., Interval Methods in Computer Graphics, Computer & Graphics, 1991, 15: 331–340.
Voiculescu, I., Berchtold, J. Bowyer, A. Martin, R. R., Zhang, Q., Interval and Affine Arithmetic for Surface Location of Power- and Bernstein-form Polynomials, The Mathematics of Surfaces IX, Eds. Cipolla R. & Martin, R. R., Springer, 2000, 410-423.
Zhang, Q., Martin, R. R., Polynomial Evaluation using Affine Arithmetic for Curve Drawing, Proceedings of Eurographics UK 2000, Eurographics UK, Abingdon, 2000, 49–56.
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Shou, H., Martin, R., Wang, G., Voiculescu, I., Bowyer, A. (2002). Affine Arithmetic and Bernstein Hull Methods for Algebraic Curve Drawing. In: Winkler, J., Niranjan, M. (eds) Uncertainty in Geometric Computations. The Springer International Series in Engineering and Computer Science, vol 704. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0813-7_12
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DOI: https://doi.org/10.1007/978-1-4615-0813-7_12
Publisher Name: Springer, Boston, MA
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