Spiral Fractals from IFS

Procedure

In general, each transformation is determined by values for r, s, θ, φ, e, and f. A method for finding these is presented in measuring IFS pieces.

Our spirals are self-similar, so for each piece r = s and θ = φ.

So we must find r, θ, e, and f.

To determine these values, select two points (a,b) and (c,d) in the fractal.

To find the parameters for the transformation giving one of the self-similar pieces of the fractal, in this piece locate the points (t,u) and (v,w) that correspond to (a,b) and (c,d).

To find r, divide the distance dist((a,b),(c,d)) = sqrt((a-c)² + (d-d)²) into dist((t,u),(v,w)). That is,

r = dist((t,u),(v,w))/dist((a,b),(c,d))

To find θ, measure the angle between the lines (a,b) to (c,d) and (t,u) to (v,w). In this example, the angle is 0.

To find e, measure the horizontal distance between (a,b) and (t,u). That is,

e = t - a.

To find f, measure the vertical distance between (a,b) and (t,u). That is,

f = u - b.

Recall that multiplying all e and f values by the same constant simply changes the scale of the picture, not the relative positions of the pieces within the picture. As a first attempt at guaranteeing the image approximately fills the window, divide all e and f values by the maximum of the vertical and horizontal extents of the fractal.

This figure shows the location of (t,u) and (v,w) for the everything else piece of this spiral. The angle between the lines (a,b) to (c,d) and (t,u) to (v,w) gives the angle theta.

Return to Spiral Fractals from IFS.