I'm using the arc-chord ratio (Du Preez, 2014) to estimate rugosity on a surface defined by a point cloud, using a Delaunay triangulation to produce the contour surface. I've noticed that the point density and surface area appear to be related by an inverse parabolic function.
Can anyone explain why this might be so? I have an intuition that it's related to fractal dimensions -- clearly, as the surface becomes more complex, its surface area increases -- but I am not a mathematician, so intuition is really all I have to go on at this point. References would be very helpful.
As it turns out, the answer is simple. If you plot the log of the parabola, you get a straight line whose slope is the fractal dimension.