L-System Fractals - Part 3

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In Part 3 of the tutorial, we are going to transform the base L-System using several complex functions.

Select the Transformation 1 properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Transformation 1

This page allows you to maintain a list of Transformations used to transform each orbit point prior to passing it to the orbit trap. The transformations are applied in series to each orbit point; i.e., the point resulting from each transformation is passed to the next transformation in the list. As you will see, this can have a dramatic effect on the resulting image. By default, there is a single transformation, Identity, and the Identity transformation does not alter the input point.

For this tutorial, we will use a single transformation.

Select the Identity properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Transformation 1
                Identity

This page is a Program Editor for a Transformation.

We want to use the Composite Function transformation so we set the Based On property to that.

Read the comments in the program's Instructions at the bottom of the page and then select the program's Properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Transformation 1
                Composite Function
                    Properties

This displays the default properties for the Composite Function transformation. The Composite Function transformation is defined by a composite function composed of 2 functions. Each function is defined via a base function and a conjugating map. For now, we will only set the F(z) and/or G(z) properties and leave the remaining properties set to their default value.

Try setting F(z) and/or G(z) to different functions and then execute the Display Fractal command on the Tools menu of the Fractal Window.

A few examples are shown below:

F(z): Recip

F(z): Csch

F(z): Bipole

F(z): Bipole, G(z): Recip

F(z): Airfoil2, G(z): Bipole

F(z): Dipole, G(z): Bipole

G(z) is set to Ident where not specified in the above examples.

Try additional functions and set some of the other properties on the page. Move up to the Composite Function transformation entry in the hierarchy and change the Based On property to try out different transformations altogether. For each transformation, select its properties page (usually called Properties) and play with the various settings that control the transformation.

When you are ready to continue with the tutorial, we need to remove the Composite Function transformation altogether. To do this, select the Transformation 1 properties page:

General
    Mandelbrot / Julia / Newton
        Orbit Trap
            Transformation 1

Now delete the selected transformation using the Delete command on the toolbar.

The transformation will be replaced by the Identity transformation.

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