Abstract
In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard affine arithmetic, and (ii) modified affine arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified affine arithmetic is not only more accurate but also much faster than standard affine arithmetic. We thus suggest that modified affine arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bowyer, A., Martin, R., Shou, H., Voiculescu, I.: Affine intervals in a CSG geometric modeller. In: Winkler, J., Niranjan, M. (eds.) Uncertainty in Geometric Computations, pp. 1–14. Kluwer Academic Publisher, Dordrecht (2002)
Bühler, K.: Linear interval estimations for parametric objects theory and application. Computer Graphics Forum 20(3), 522–531 (2001)
Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. Anais do VII SIBGRAPI, 9–18 (1993), Available at http://www.dcc.unicamp.br/~stolfi/
De Cusatis Jr., A., De Figueiredo, L.H., Gattass, M.: Interval Methods for Ray Casting Implicit Surfaces with Affine Arithmetic. In: XII Brazilian Symposium on Computer Graphics and Image Processing, pp. 65–71 (1999)
De Figueiredo, L.H.: Surface intersection using affine arithmetic. In: Proceedings of Graphics Interface, pp. 168–175 (1996)
De Figueiredo, L.H., Stolfi, J.: Adaptive enumeration of implicit surfaces with affine arithmetic. Computer Graphics Forum 15(5), 287–296 (1996)
Heidrich, W., Seidel, H.P.: Ray tracing procedural displacement shaders. In: Proceedings of Graphics Interface, pp. 8–16 (1998)
Heidrich, W., Slusallek, P., Seidel, H.P.: Sampling of procedural shaders using affine arithmetic. ACM Trans. on Graphics 17(3), 158–176 (1998)
Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G.: Comparison of interval methods for plotting algebraic curves. Computer Aided Geometric Design 19(7), 553–587 (2002)
Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. Ellis Horwood (1984)
Shou, H., Martin, R., Voiculescu, I., Bowyer, A., Wang, G.: Affine arithmetic in matrix form for polynomial evaluation and algebraic curve drawing. Progress in Natural Science 12(1), 77–80 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shou, H., Lin, H., Martin, R., Wang, G. (2003). Modified Affine Arithmetic Is More Accurate than Centered Interval Arithmetic or Affine Arithmetic. In: Wilson, M.J., Martin, R.R. (eds) Mathematics of Surfaces. Lecture Notes in Computer Science, vol 2768. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39422-8_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-39422-8_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20053-6
Online ISBN: 978-3-540-39422-8
eBook Packages: Springer Book Archive