1995/6 Research Journal
Concurrent Computation Group
M. E. Montiel and E. J. Zaluska
Fractals have been recognised as a powerful description of many natural objects for several years[1]. Their definition is generally based on concepts of dimension which reflect the fragmentation or irregularity feature of these structures[2]. In this work we investigate the definition of fractals by extending geometric descriptions to multiple resolutions. Based on this approach we propose a fractal definition as a composition of exponential functions embedded in a multidimensional domain. By considering transformations between different resolutions we provide a definition of fractals capable of modelling continuous deformations.
The aim of modelling these deformations is to extend fractal descriptions to animated forms.
If objects are represented by sets then a geometric mapping can be characterised by a function f which assigns 
for each 
in such a way that the geometrical properties of the sets are left intact
Each geometry is defined by a particular group of mappings which are called a group of motions[3]. Isometries, for example, are a group of motions which preserve distance between points and they characterise Euclidean geometry. Affine geometry is defined when transformations of scale are included. Other geometries such as inversive, differential and topology are defined by more complex mappings. Fractal geometry is characterised by a group of motions applied to different resolutions within the same object. That is, equation (1) is represented by
where the subindex i and j represent two resolutions of an object X. According to this definition it is possible to apply the mappings of different geometries to create a fractal. In the literature the creation of fractals has been restricted to the use of the mappings of Euclidean and affine geometry[4]. In this work we propose to extend the mappings to include homeomorphism. These mappings characterise topology which can be defined as the geometry of continuous deformation.
The recursive mappings in fractals are characterised by a functional equation which describes a set of transformations applied to an original set. Particular characterisations of this function are the formulation of Iterative Function Systems (IFS)[5] where fractals are created by the recursive use of matrix transformations and grammars applied to descriptions of lines[6]. Dubuc[7] considers an analytic form of a functional equation which defines irregular curves and whose formulation corresponds to a parametric complex-valued function. Here, we extend this definition to a multidimensional parametric space. This extension allows curves with a self-similar property to be created. We consider the composition of a fractal F as a linear additive combination of functions which describe different resolutions
According to equation (2) each function in this equation has a recursive definition. That is
Therefore, an object is described by a composite map of the form
. The function f can be characterised by the motions of any geometry. If f is a homeomorphism (i.e. a continuous one-to-one mapping of the plane onto itself) then the resolutions of an object are related by general mappings.
In order to develop equation (3) by a homeomorphic mapping it is necessary to specify a level of detail 
using a general function. There exist many potential ways of describing this function, we use a parameterised decomposition based on a series of sinusoids at different frequencies. That is,
where the coefficients 
and 
correspond to two orthonormal vectors
This equation defines a mapping from an i-dimensional space to the complex plane, and it can be proven that it corresponds to the generalisation of the mappings described in affine geometry to functions represented by different frequencies.
Figure 1: Example of a fractal obtained by a change in phase
The combination of equations (3) and (5) defines a mathematical representation which allows several irregular curves to be created. We have created several fractals by defining the domain of the function as a discrete set of points. As an example, consider the structure defined by only one term in the summation of equation (5) with 
, then
Figure 2: Sequence of a fractal animation
Figure 1 shows the result of evaluating this function with a
domain 
specified by [0,2 
/3,4 
/3] and a recursive definition of the values of 
and 
given by
and 
.
A dynamic fractal can be created by parameterising equation (5) in time. This parameterisation corresponds to a function 
which describes the deformation of a complete structure in therms of transformations in each resolution. These transformations define a change in a resolution 
until it occupies the space defined by another function 
. That is
These kind of functions can be formalized mathematically through the mappings defined in topology and they can be used to model continuous deformations of fractals.
Figure 2 shows a sequence of a fractal deformation produced by changing the definition of the phase in each resolution as a function in time. It can be seen from the figure that simple changes can produce a complex dynamic behaviour where one cannot discover the rules of articulation and where geometric patterns emerge forming a fragmented shape.
In this work we have considered the extension of the patterns defined in fractal geometry to dynamic descriptions. In the same way in which irregular curves have been applied to the study and modelling of complex forms such as plants, trees, molecular chains, clouds, shells, ice formation or ocean waves[1, 5, 8], the extension presented here permits the analysis and synthesis of models for the continuous distortion of complex forms. The extension is developed using a functional representation based on mappings which define levels of detail through geometric transformations. These transformations are parameterised in time to obtain dynamic structures.
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Copyright (c) 1996 University of Southampton, June 1996.
