All Bezier curves are attractors of iterated function systems (original) (raw)
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Many fractal curves can be produced as the limit of a sequence of polygonal curves, where the curves are generated via an iterative process, for example an L-system. One can visualise such a sequences of curves as an animation that steps through the sequence. A small part of the curve at one step of the iteration is close to a corresponding part of the curve at the previous step, and so it is natural to add frames to our animation that continuously interpolate between the curves of the iteration. We introduce sculptural forms based on replacing the time dimension of such an animation with a space dimension, producing a surface. The distances between the steps of the sequence are scaled exponentially, so that self-similarity of the curves is reflected in self-similarity of the surface. For surfaces based on the constructions of certain fractal curves, the approximating polygonal curves self-intersect, which means that the resulting surface would also self-intersect. To fix this we smooth the polygonal curves. We outline two very different approaches to producing and smoothing the geometry of such a "developing fractal curve" surface, one based on a direct parameterisation, and the other based on Loop subdivision of a coarse polygonal control mesh. *
Construction of fractal objects with iterated function systems
ACM SIGGRAPH Computer Graphics, 1985
In computer graphics, geometric modeling of complex objects is a difficult process. An important class of complex objects arise from natural phenomena: trees, plants, clouds, mountains, etc. Researchers are at present investigating a variety of techniques for extending modeling capabilities to include these as well as other classes. One mathematical concept that appears to have significant potential for this is fractals. Much interest currently exists in the general scientific community in using fractals as a model of complex natural phenomena. However, only a few methods for generating fractal sets are known. We have been involved in the development of a new approach to computing fractals. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique "attractor" which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design...
Developments in fractal geometry
Bulletin of Mathematical Sciences, 2013
Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.
Linear Fractal Shape Interpolation
Interpolation of two-dimensional shapes described by iterated function systems is explored. Iterated function systems define shapes using self-transformations, and interpolation of these shapes requires interpolation of these transformations. Polar decomposition is used to avoid singular intermediate transformations and to better simulate articulated motion. Unlike some other representations, such as polygons, shaped described by iterated function systems can become totally disconnected. A new, fast and image-based technique for determining the connectedness of an iterated function system attractor is introduced. For each shape interpolation, a two parameter family of iterated function systems is defined, and a connectedness locus for these shapes is plotted, to maintain connectedness during the interpolation.
Self-affine Fractals Generated by Nonlinear Systems
2008
A system of ODE’s with nonlinear terms exhibits a nonlinear dynamic behavior. Under some conditions these terms can be locally approximated by linear factors, which can be, after discretization transformed in the sequence of (hyperbolic) Iterated Function Systems (IFS) that generates a unique fractal attractor. This attractor reflects the dynamics in the vicinity of the approximated point of the nonlinear system. Here, the IFS is replaced with an associate AIFS (Affine invariant IFS), a kind of IFS that has affine invariance property and permits further manipulating of this fractal attractor.
Exploring Rational Bezier Curves through Iterated Function Systems
2010
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.
On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces
2020
In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function.
Fast Visualisation and Interactive Design of Deterministic Fractals
2008
This paper describes an interactive software tool for the visualisation and the design of artistic fractal images. The software (called AttractOrAnalyst) implements a fast algorithm for the visualisation of basins of attraction of iterated function systems, many of which show fractal properties. It also presents an intuitive technique for fractal shape exploration. Interactive visualisation of fractals allows that parameter changes can be applied at run time. This enables real-time fractal animation. Moreover, an extended analysis of the discrete dynamical systems used to generate the fractal is possible. For a fast exploration of different fractal shapes, a procedure for the automatic generation of bifurcation sets, the generalizations of the Mandelbrot set, is implemented. This technique helps greatly in the design of fractal images. A number of application examples proves the usefulness of the approach, and the paper shows that, put into an interactive context, new applications of these fascinating objects become possible. The images presented show that the developed tool can be very useful for artistic work.
CONSTRUCTION OF SMOOTH FRACTAL SURFACES USING HERMITE FRACTAL INTERPOLATION FUNCTIONS
This paper approaches the Hermite interpolation problem using fractal interpolation procedures. We generalise some theorems provided by Barnsley and others regarding the differentiability of fractal interpolation functions, when recurrent iterated function systems are used. In addition, we generalise the construction given by Navascués and Sebastián in [11] and we provide a construction of smooth (C 1 ) fractal surfaces using C 1 Hermite Fractal Interpolation Functions.