Exploring Rational Bezier Curves through Iterated Function Systems Bhagwati Prasad and Kuldip Katiyar Department of Mathematics Jaypee Institute of Information Technology University A-10, Sector-62, Noida, UP-201307 INDIA e-mail: b_prasad10@yahoo.com ; bhagwati.prasad@jiit.ac.in Abstract— Bezier and rational Bezier curves are important elements in computer aided geometric design (CAGD) due to their capability to represent both the free-form setting and the algebraic one as well. Therefore, such curves are popularly accepted as a standard representation for designing problems. The purpose of this paper is to present the rational Bezier curves as attractors of some iterated function systems. Keywords-Fractals; de Casteljau subdivision algorithm; rational Bezier curve; Iterated function system. I. INTRODUCTION Mandelbrot introduced fractal as a set for which the Hausdorff dimension strictly exceeds the topological dimension. He realized that fractal geometry is a powerful tool to characterize intrinsically irregular system found in nature such as trees, clouds, mountains and lightnings etc having in common the property that if one magnifies a small portion of them, a complexity comparable to that of the entire structure is revealed (see Mandelbrot [11] and Falconer [4]). Fractals are characterized by their property of self similarity. Barnsley [1] used this property in order to encode the different parts of the fractal by contractive operators. The set of these operators in a complete metric space form an Iterated Function System (IFS). These IFS were first studied by Hutchinson [9] in a pure mathematical context. Barnsley [1]-[3], further developed this tool in fractal geometry and computer graphics. The role of curves and surfaces in computer aided geometric design is well illustrated in [5]-[7], [12], [13] and several references thereof. For details regarding fractals and IFSs, one can see Barnsley [1]-[3], Falconer [4] and Hutchinson [9]. The IFSs technique is a popular approach for modelling fractals which transforms any initial shape into a union of its reduced copies. The contraction property of the operators is used to prove the existence of a unique invariant set named the attractor. Recently, Goldman [8] presented a constructive procedure based on the de Casteljau subdivision algorithm for Bezier curves that builds an IFS whose attractor is exactly the given Bezier curve. Here we shall extend his approach to rational Bezier curves. II. PRELIMINARIES In this section we present the basic definitions and concepts required for our results. Definition 1.1[3]. Let (X, d) be a metric space. A transformation f: X ⟶ X is said to be Lipschitz with Lipschitz constant L ∈ R iff d(f(x), f(y)) ≤ L d(x, y) for all x, y ∈ X. A transformation f: X ⟶ X is called contractive iff it is Lipschitz with Lipschitz constant L ∈ [0, 1). A Lipschitz constant L ∈ [0, 1) is also called a contraction factor. Definition 1.2[1]. A hyperbolic iterated function system (IFS) consists of a complete metric space (X, d) together with a finite set of contraction mappings f r : X ⟶ X, with respective contractivity factors s r for r = 1, 2, ..., n. This IFS is represented by {X; f r : r = 1, 2, ..., n} with contractivity factor s = max{s r : r = 1, 2, ..., n}. Definition 1.3[3]. Let (X, d) be a metric space and H(X) denote the nonempty compact subsets of X. Then the Hausdorff metric h in H(X) is defined as h(A, B) = max {d(A, B), d(B, A)} for all A, B ∈ H(X), where d(A, B) = max(min(d(a, b): b ∈ B): a ∈ A). Theorem 1.4 [1]. Let {X; w r , r = 1, 2, ..., n} be a hyperbolic iterated function system with contractivity factor s. Then the transformation W: H(X) ⟶ H(X) defined by ) ( ) ( 1 B w B W n r r  1  for all B ∈ H(X) is a contraction mapping on the complete metric space (H(X), h) with contractivity factor s. That is, h(W(B), W(C)) ≤ s h(B, C) for all B, C ∈ H(X). Its unique fixed point, A ∈ H(X) obeys ) ( ) ( 1 A w A W A n r r  1  W and is given by ) ( lim B W A r r W l for any B ∈ H(X). This theorem ensures the existence of a unique fixed point also called an attractor of the IFS (see [1] - [3]). III. RATIONAL BEZIER CURVE AND SUBDIVISION SCHEMES We follows Farin [5]-[6], Goldman [7]-[8], John [10] and Piegl [12] for the notations, definitions and concepts developed in this section. A rational Bezier curve R(r) of degree n is defined as n j j n j n j j j n j w r B w P r B r R 0 0 ) ( ) ( ) ( for 0 ≤ r ≤ 1 where P 0 , P 1 , ..., P n Third International Conference on Emerging Trends in Engineering and Technology 978-0-7695-4246-1/10 $26.00 © 2010 IEEE DOI 10.1109/ICETET.2010.73 267 Third International Conference on Emerging Trends in Engineering and Technology 978-0-7695-4246-1/10 $26.00 © 2010 IEEE DOI 10.1109/ICETET.2010.73 267 Third International Conference on Emerging Trends in Engineering and Technology 978-0-7695-4246-1/10 $26.00 © 2010 IEEE DOI 10.1109/ICETET.2010.73 267 Third International Conference on Emerging Trends in Engineering and Technology 978-0-7695-4246-1/10 $26.00 © 2010 IEEE DOI 10.1109/ICETET.2010.73 267