Authors:
J. F. Sanjuan-Estrada
;
L. G. Casado
and
I. Garćıa
Affiliation:
University of Almeria, Spain
Keyword(s):
Geometric computing, Roots of complex polynomials, Interval arithmetic, Rendering complex space, Ray tracing.
Related
Ontology
Subjects/Areas/Topics:
Computational Geometry
;
Computer Vision, Visualization and Computer Graphics
;
Image Formation and Preprocessing
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.
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