| Suppose we want to build a
unifractal cartoon with H = 0.25 and dt1 = 0.3. |
| The scaling condition implies
dt2 is determined by the equation |
| 0.30.25 - (dt2)0.25 +
(1 - 0.3 - dt2)0.25 = 1. |
| (Recall dt3 is completely specified by
dt1 and dt2.) |
| Mathematica can find dt2 using the
command |
| FindRoot[0.3^0.25 + x^0.25 + (1 - 0.3 - x)^0.25 == 1,{x,2}] |
| This gives |
| dt2 = x = 0.135608 |
| Then from dt1 + dt2 + dt3 = 1, we obtain |
| dt3 = 1 - 0.3 - dt2 = 0.564392. |
| We calculate the
dYi from these by the unifractal scaling relation
|dYi| = (dti)H, and the up, down, up ordering of the
generator segments: |
| dY1 = 0.30.25 = 0.740083 |
| dY2 = -(0.1356080.25) = -0.606836, and |
| dY3 = (0.564392)0.25 = 0.86675. |
|
| The turning points are |
| (a, b) = (dt1, dY1) = (0.3, 0.740083), and |
| (c, d) = (dt1 + dt2, dY1 + dY2) = (0.435608, 0.133247). |
|
| The generator is on the left; the 8th iterate, sampled at 3128 points, is on the right. |
|
