A new construction of fractal interpolation surfaces, using solutions of partial differential equations, is presented. We consider a set of interpolation points placed on a rectangular grid and a specific PDE, such that its Dirichlet's problem is uniquely solvable inside any given orthogonal region. We solve the PDE, using numerical methods, for a number of regions, to construct two functions

Fractals are known to produce extremely complicated and impressive shapes, which many times resemble objects of the physical world. They have been used extensively in various scientific areas such as computer graphics, image compression and processing, biology and metallurgy. They usually emerge as attractors of Iterated Function Systems, a notion introduced by Barnsley and Demko in [

In this paper, we will deal with the construction of fractal surfaces, a subject firstly addressed by Massopust in [

In the years that followed, the problem of the construction of fractal surfaces has been dealt by many other authors (see [

As mentioned above, an

Two known attractors that arise from IFS. (a) Sierpinski's Triangle, (b) a fern.

There are two known algorithms, that are used to construct the attractor of an IFS, the Deterministic Algorithm (DA) and the Random Iteration Algorithm (RIA). The first starts with an arbitrary compact set

A more general concept, that allows the construction of even more complicated sets, is that of the

In this case, the construction of the contractive map

Next, we construct a sequence of sets that converge to the attractor. Let

The attractors of the IFS defined by (

Both DA and RIA can be modified to construct the attractor of any RIFS. The modification of DA is described above, where the sequences

In this section, we present a general construction of fractal surfaces, that interpolate points placed on a rectangular grid. The fractal surfaces emerge as attractors of specific RIFS. Consider a data set:

such that

such that

The interpolation points divide

To complete the definition of the RIFS, we consider

for all

Consequently, the RIFS

Let

The proof of the above proposition, together with several methods that one may use to select suitable functions

Let

The functions

It is well known that in the case of the construction of affine FIFs of one variable, the attractor depends only on the choice of the interpolation points (and of course on the choice of the vertical scaling factors). It was shown in [

find a function

find a function

where

Consider the set

Now, consider a function

We summarize the procedure of the construction.

We take (as input) a set

We select a set

We construct arbitrary

We solve (numerically) the partial differential equation inside the region

We solve (numerically) the partial differential equation inside

Finally, we construct the attractor of the corresponding RIFS (see Figure

An example of the construction. Here

In Section

An example of the algorithm, where

INPUT: The

The initial set

The projections of the interpolation points on

Form the function H.For

For

We solve numerically, inside the region

Form the function B. For

For

We solve numerically, inside the domain

Form the attractor of the RIFS. For

For

For

We deal with the points of the region

As the parameters of the map

The new set G contains all the points that were produced by the previous iterations (

OUTPUT: After the completion of the iterations, the final set

As noted, the output of the algorithm is a set of points, that their projections on

The two fractal surfaces (shown above) were constructed using the same parameters, but different partial differential equations. In (a) Laplace's equation was used (

Some examples of the proposed construction. At the left side, the interpolation points and the chosen borders are shown. At the right side, one can see the constructed fractal surfaces. We used Laplace's PDE in both examples.

The author would like to thank Professor M. Drakopoulos (Mathematics Department of the University of Athens) for his valuable help in producing the algorithms that solve Dirichlet's problem for Lapalce's PDE. This reserch is partially supported by the Special Account for Research Grants of the University of Athens no. 70/4/5626.