# RENDERING (COMPLEX) ALGEBRAIC SURFACES

### J. F. Sanjuan-Estrada, L. G. Casado, I. Garćıa

#### Abstract

The traditional ray-tracing technique based on a ray-surface intersection is reduced to a surface-surface intersection problem. At the core of every ray-tracing program is the fundamental question of detecting the intersecting point(s) of a ray and a surface. Usually, these applications involve computation and manipulation of non-linear algebraic primitives, where these primitives are represented using real numbers and polynomial equations. But the fast algorithms used for real polynomial surfaces are not useful to render complex polynomials. In this paper, we propose to extend the traditional ray-tracing technique to detect the intersecting points of a ray and complex polynomials. Each polynomial equation with some complex coefficients are called complex polynomials. We use a root finder algorithm based on interval arithmetic which computes verified enclosures of the roots of a complex polynomial by enclosing the zeros in narrow bounds. We also propose a new procedure to render real or complex polynomials in the real and the complex space. If we want to render a surface in the complex space, the algorithm must detect all real and complex roots. The color of a pixel will be calculated with those roots with an argument inside a selected complex space and minimum magnitude of the complex roots.

#### References

- Glassner, A. (1989). An Introduction to Ray Tracing. Academic Press, Boston.
- Granas, A. and Dugundji, J. (2004). Fixed point theory. Bulletin of the American Mathematical Society, 41(2):267-271.
- 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.
- Jimenez-Melado, A. and Morales., C. (2005). Fixed point theorems under the interior condition. Procceding of the American Mathematical Society, 134(2):501-507.
- Kalantari, B. (2004). Polynomiography and applications in art, education, and science. Computers & Graphics, 28(3):417-430.
- Whitted, T. (1980). An improved illumination model for shaded display. Commun. ACM, 23(6):343-349.

#### Paper Citation

#### in Harvard Style

F. Sanjuan-Estrada J., G. Casado L. and Garćıa I. (2006). **RENDERING (COMPLEX) ALGEBRAIC SURFACES** . In *Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,* ISBN 972-8865-40-6, pages 139-146. DOI: 10.5220/0001369901390146

#### in Bibtex Style

@conference{visapp06,

author={J. F. Sanjuan-Estrada and L. G. Casado and I. Garćıa},

title={RENDERING (COMPLEX) ALGEBRAIC SURFACES},

booktitle={Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,},

year={2006},

pages={139-146},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001369901390146},

isbn={972-8865-40-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,

TI - RENDERING (COMPLEX) ALGEBRAIC SURFACES

SN - 972-8865-40-6

AU - F. Sanjuan-Estrada J.

AU - G. Casado L.

AU - Garćıa I.

PY - 2006

SP - 139

EP - 146

DO - 10.5220/0001369901390146