Keyword(s):Computational geometry, Complex roots of real polynomials, Video sequences in complex space.

Related
Ontology
Subjects/Areas/Topics:Computer Vision, Visualization and Computer Graphics
;
Geometric Computing
;
Geometry and Modeling
;
Rendering
;
Rendering Algorithms

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.(More)

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.

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F. Sanjuan-Estrada J.; G. Casado L.; García I. and (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

@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}, }

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