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 Title: Subdivision Matrices and Iterated Function Systems for Parametric Interval Bezier Curves Author(s): O. Ismail Pages: 1-6 Paper ID:170602-9494-IJVIPNS-IJENS Published: April, 2017 Abstract: Fractals are famous both for their strange appearance and for their odd geometric properties. The problem of modeling is very simple when one has mathematical description of the fractal he wants to model. A set of transformations that generates a fractal by iteration is called an iterated function systems (IFS). An iterated function system maps the corresponding fractal onto itself as a collection of smaller self-similar copies. Fractals are often deﬁned as ﬁxed points of iterated function systems because when applied to the fractal the transformations that generate a fractal do not alter the fractal. In this paper we are going to show that the parametric interval Bezier curves are indeed fractals if the four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original interval Bezier curves are fractals. The four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original parametric interval Bezier curve are obtained. Iterated function systems (IFS) for the four fixed Kharitonov's polynomials (four fixed Bezier curves) are constructed so that starting with any compact set, not just the four fixed Kharitonov's polynomials (four fixed Bezier curves) control polygons, and iterating the transformations, the resulting sets converge in the limit to the given four fixed Kharitonov's polynomials (four fixed Bezier curves) curves. The transformation matrices in the iterated function systems are constructed to mimic the subdivision procedure, so there is indeed a deep connection between subdivision algorithms and fractal procedures. The fixed control points that subdivide the four fixed Kharitonov's polynomials (four fixed Bezier curves) curves into two four fixed Kharitonov's polynomials (four fixed Bezier curves) curves are afﬁne combinations of the original four fixed Kharitonov's polynomials (four fixed Bezier curves) control points. Thus we can represent the four fixed Kharitonov's polynomials (four fixed Bezier curves) subdivision by two square matrices whose entries are the coefﬁcients in these afﬁne combinations. A numerical example is included in order to demonstrate the effectiveness of the proposed method. Keywords: Subdivision matrices, iterated function systems, interval Bezier curve, CAGD. Full Text (.pdf) | 565 KB