RELIABLE COMPUTATION OF ROOTS TO RENDER REAL POLYNOMIALS IN COMPLEX SPACE

J. F. Sanjuan-Estrada, L. G. Casado, I. García

2006

Abstract

Many geometric applications involve computation and manipulation of non-linear algebraic primitives. These basic primitives like points, curves and surfaces are represented using real numbers and polynomial equations. For example, ray tracing technique rendering three-dimensional realistic images, where each pixel need to find the minimum positive root of intersection point when a lineal ray hit a surface. However, the intersection between a ray and a polynomial equation has differents roots, where each root can be a real number (without imaginary part) or a complex number (with real and imaginary part), so that, the number of roots is equal to degree of polynomial. In this paper, we extend the traditional ray tracing technique to show roots in the complex space. We use an algorithm that analyse all verified roots of intersection point using interval arithmetic. This algorithm computes verified enclosures of the roots of a polynomial by enclosing the zeros in narrow bounds. The reliability of the algorithm depends on the accurate evaluation of these complex roots. Finally, we propose differents solutions to render a image in the complex space, where the arguments of complex roots are used to choose the roots of intersection point in complex space, while the color of each pixel is computed by minimum modulus of complex roots chosen.

References

  1. C. Sidney Burrus, J.W. Fox, G. S. and S.Treitel. (2003). Horner's method for evaluating and delating polynomials. Rick University.
  2. Georg, S. (1990). Two methods for the verified inclusion of zeros of complex polynomials. In Ullrich, C., editor, Contributions to computer arithmetic and sefl-validating numerical methods , pages 229-244. IMACS, Scientific Publishing Co.
  3. Granas, A. and Dugundji, J. (2004). Fixed point theory. Bulletin of the American Mathematical Society, 41(2):267-271.
  4. Gruner, K. (1987). Solving complex problems for polynomials and linear systems with verified high accuracy. In Kaucher, E., K. U. and Ullrich, C., editors, Computer arithmetic, scientific computation and programming languages, pages 199-220.
  5. Hammer, R., Hocks, M., Kulisch, U., and Ratz, D. (1995). C++ Toolbox for Verified Computing I: Basic Numerical Problems: Theory, Algorithms, and Programs. Springer-Verlag, Berlin.
  6. Jimenez-Melado, A. and Morales., C. (2005). Fixed point theorems under the interior condition. Procceding of the American Mathematical Society, 134(2):501-507.
  7. Smith, B. T. (1970). Error bounds for zeros of a polynomial based upon Gerschgorin's theorems. Journal of the ACM, 17(4):661-674.
Download


Paper Citation


in Harvard Style

F. Sanjuan-Estrada J., G. Casado L. and García I. (2006). RELIABLE COMPUTATION OF ROOTS TO RENDER REAL POLYNOMIALS IN COMPLEX SPACE . In Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, ISBN 972-8865-39-2, pages 305-312. DOI: 10.5220/0001355603050312


in Bibtex Style

@conference{grapp06,
author={J. F. Sanjuan-Estrada and L. G. Casado and I. García},
title={RELIABLE COMPUTATION OF ROOTS TO RENDER REAL POLYNOMIALS IN COMPLEX SPACE},
booktitle={Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},
year={2006},
pages={305-312},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001355603050312},
isbn={972-8865-39-2},
}


in EndNote Style

TY - CONF
JO - Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,
TI - RELIABLE COMPUTATION OF ROOTS TO RENDER REAL POLYNOMIALS IN COMPLEX SPACE
SN - 972-8865-39-2
AU - F. Sanjuan-Estrada J.
AU - G. Casado L.
AU - García I.
PY - 2006
SP - 305
EP - 312
DO - 10.5220/0001355603050312