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Transformation

The images under enlargement and translation are similar to the original object. The same is true for reflection and rotation. These transformations are called similarity transformations. They preserve the shape but not necessarily the size. Corresponding angles are the same in both the object and the image whereas corresponding line segments must all have the same scale factor of proportionality.

Similarity transformations are special cases of affine transformations which map parallelograms to parallelograms. Under an affine transformation, a square can be transformed into a rectangle while a triangle can be transformed into a triangle with different angles. Geometrically affine transformations include shearing as well as scaling (enlargement), reflection, rotation and translation. The following figure shows a shape which is composed of 4 self-affine copies.

The two-branch tree below consists of two reduced copies of the whole and the stem. The stem is not similar to the whole tree but is the image of an affine transformation which compresses the tree into a line. The tree is self-affine instead of self-similar.

Fractals can be generated by IFS through repeated application of affine transformations, provided that they are contractive i.e. any two points in the object will be brought closer together under the corresponding transformation.

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