Transformations and Base fields

 

Transformations:

Transformations used to create fractal structures: Most of the self-similar fractal structures are created with the help of a number of transformations (i.e. 'function of points' or 'vector functions' which are such functions which accepts the coordinates of a point and returns coordinates of another point as the functional value). A fractal structure consists of the set of points which when put into a transformation that creates it returns a point that lies in the fractal set (or fractal structure) itself. Thus the fractal set remains invariant under transformations that create it.

Transformations used to create fractal structures with FC 2.0: FC 2.0 can make fractal structures which are made up of linear transformations i.e. transformations of the form T(x , y) = (ax + by + c , px + qy + r) . These forms of transformations have very easy physical interpretation. Such a transformation can be split up into a combination of a contraction (or enlargement), a rotation and a translation. Thus, T(x , y) = Translation( Rotation( Contraction(x , y)))

But, how to determine transformations used in a fractal structure? We first imagine a fractal structure. Then, we find out the different self-similar parts of it. Then we are to determine mathematically what contraction (or enlargement), rotation and translation when applied to a point on the main structure creates the corresponding point on the said part of the structure. The combination of contraction (or enlargement), rotation and translation hence determined constitutes the transformation. The following simple example will make it clear:

For example we want to create a leaf:

The main structure is shown within the red boundary.

To create the left part (in pink boundary) we apply a transformation on the main leaf. The transformation applied are a contraction of 0.5 along both X and Y, followed by a rotation of 30 degrees in anti-clockwise direction (i.e.+30), followed by a translation of 12 units along Y axis.

Thus, we produced the left part of the leaf.

Similarly, to create the right part of the leaf we apply a contraction of 0.45 along both X and Y, followed by a rotation of 40 degrees in clockwise direction (i.e.-40), followed by a translation of 11 units along Y axis.

To create the top part of the leaf we apply a contraction of 0.6 along both X and Y, followed by a rotation of 10 degrees in clockwise direction (i.e.-10), followed by a translation of 20 units along Y axis

To understand the transformations in details, try by experimenting with the supplied fractal structure files.

 

Base Field:

Base fields are easy to understand. It is the collection of points on which transformations are applied. In the above example of the leaf the base field is the stem of the leaf. To make it clear, note the changes made to the leaf by changing the base field.

 

In the first leaf some horizontal dashes are added to the base field (stem). In the second leaf upper half of the stem is removed.