# Comparing Measures of Line Jaggedness

Comparing Measures of Line Jaggedness

This Demonstration compares three statistical measures of line jaggedness visually and numerically. It creates and plots rugged curves with the Mandelbrot–Weierstrass function, , of a chosen fractal dimension , starting point, range, and resolution, and curve-fits the data with a third- or fourth-degree polynomial. The fitted ("smoothed") curve is superimposed on the generated curve with the residuals plotted below. The residuals' standard deviation, , and two additional jaggedness indices, and , having a range from 0 (perfectly smooth) to 1, are calculated and their values displayed. Also included is a plot showing these two indices' dependence on for a chosen range factor .

MW(t)=1/b1-kcosbt+ϕ

T

∑

n=1

n(2-d)

n

d

MW(t)

σ

j(σ)

2

j(σ)

3

σ

c