500 ≤ 2000 ≤ 50000
|
A(p) = M(p) + v
where M is a 2 X 2 matrix (the linear part) and v is a 2 X 1 column vector (the translation part). Here p is a point in the plane represented as a 2 X 1 column vector. In terms of components, if p = (x0, y0), then (x1, y1) = A(p) is expressed byx1 = m11*x0+m12*y0 + v1 |
y1 = m21*x0+m22*y0 + v2 |
Note: Affine transformations with determinant zero are assigned small probabilities to guarantee that they will be selected.
For example, the Sierpinski fractal is generated by the three affine transforms
A1: | |||
x1 = 0.5x0+0.0y0 + 0.0 | |||
y1 = 0.0x0+0.5y0 + 0.0 |
A2: | |||
x1 = 0.5x0+0.0y0 + 0.5 | |||
y1 = 0.0x0+0.5y0 + 0.0 |
A3: | |||
x1 = 0.5x0+0.0y0 + 0.250 | |||
y1 = 0.0x0+0.5y0 + 0.433 |
Similarly, the Fern fractal is generated by using four affine transformation including A4
A4: | |||
x1 = 0.0x0+0.0y0 + 0.4987 | |||
y1 = 0.0x0+0.3y0 + 0.0070 |
For more details, see Introduction to Fractals and Chaos, Richard M. Crownover, Jones and Bartlett, 1995.