Abstract: Iterated coordinate transformation systems (ICTS) are a family of iterated function systems (IFS) based on simple coordinate transformations. Images of ICTS are created by repeatedly applying a specified mathematical function to a set of data or image. ICTS exhibit many of the hallmarks of iterated systems including a tendency to produce spiral and self-similar patterns.

I. Introduction

II. Some Classes of ICTS

III. Creating Images of ICTS

IV. Non-repeating Transform Sequences (paper)

IV. Non-repeating Transform Sequences (Images)

Figure 1: Form 0 - The 'Polar Coordinates' transform.

Xnew = cos(Xcur)*Ycur ; Ynew = sin(Xcur)*Ycur

[This is perhaps the most well known of the entire family of ICTS; as it is built into Adobe Photoshop. This shape will be produced by repeated application of the 'Polar Coordinates' filter.]

Introduction: Iterated function systems (IFS) are created by the repeated mapping of a set of points in the plane to a new set of locations. This mapping is accomplished by computing new (X,Y) coordinates for a point using it's current (X,Y) coordinates as values in some function :
Xnew = f(Xcur,Ycur) ; Ynew = f'(Xcur,Ycur).
The resultant new (X,Y) become the current (X,Y) values for the next iteration of the function.

Iterated coordinate transformation systems (ICTS) are a subset of IFS in which the mappings are limited to elementary coordinate transformations. The elementary coordinate transformations are used as primitives to build up a coordinate transformation function. This function is then applied repeatedly to the starting image. The starting image has little impact on the final form of the computed ICTS if several iterations are carried out; however in cases where only a few iterations are calculated the starting data set or image will strongly influence the final image.

One well known coordinate transformation is the polar coordinate transformation. This operation converts lines paralell to the x-axis to circles centered on the origin, and lines paralell to the y-axis to rays emanating from the origin. The goal of this work is to explore the artistic value of using this polar coordinate transformation in combination with other elementary transformations in ICT systems. All the coordinate transformations used here are continuous functions. Thus proximal points in the data set remain proimal. I believe this to be an important consideration in the aesthetic quality of the results. This family of function systems is explored herein and examples of many of the major sub-classes are shown.

Some Classes of ICTS:
ICTS can be grouped according to the elementary transformations making up the overall coordinate transformation. Form 0 highlights the attractor of the polar transform; a double spiral with on mirror plane of symmetry. This Form represents the starting point for this exploration of ICTS.

Form 6 is a representative sample of a class of two element ICTS. First the polar transform, followed by a 45 degree clockwise rotation. This Form shows the effects of symmetry breaking; the 45 degree rotation moves 25% of the image across the mirror symmetry plane created by the polar transform. Form 6 also highlights one of the hallmarks of IFS - the self-similar spiral. These ICTS are not chaotic with respect to the trasformations combined to create them. Shown here are several members of the Polar > Rotate ICTS class.

Figure 2: Form 6 - Polar > Rotate 45
Xnew = cos(Xcur)*Ycur ; Ynew = sin(Xcur)*Ycur ;
Rotate image 45 degrees

Figure 3: Form 14 - Polar > Rotate 45 > Flip Vertical

III. Creating Images of ICTS