Tortuosity

Tortuosity is a property of curve being (twisted; having many turns). There have been several attempts to quantify this property. Tortuosity is commonly used to describe diffusion in porous media, such as soils and snow.

Subjective estimation (sometimes aided by optometric grading scales) is often used.

The simplest mathematical method to estimate tortuosity is the arc-chord ratio: the ratio of the length of the curve (L) to the distance between the ends of it (C):

Arc-chord ratio equals 1 for a straight line and is infinite for a circle.

Another method, proposed in 1999, is to estimate the tortuosity as the integral of the square (or module) of the curvature. Dividing the result by length of curve or chord has also been tried.

In 2002 several Italian scientists proposed one more method. At first, the curve is divided into several (N) parts with constant sign of curvature (using hysteresis to decrease sensitivity to noise). Then the arc-chord ratio for each part is found and the tortuosity is estimated by:

In this case tortuosity of both straight line and circle is estimated to be 0.

In 1993 Swiss mathematician Martin Mächler proposed an analogy: it’s relatively easy to drive a bicycle or a car in a trajectory with a constant curvature (an arc of a circle), but it’s much harder to drive where curvature changes. This would imply that roughness (or tortuosity) could be measured by relative change of curvature. In this case the proposed "local" measure was derivative of logarithm of curvature:

However, in this case tortuosity of a straight line is left undefined.

In 2005 it was proposed to measure tortuosity by an integral of square of derivative of curvature, divided by the length of a curve:

In this case tortuosity of both straight line and circle is estimated to be 0.

Fractal dimension has been used to quantify tortuosity. The fractal dimension in 2D for a straight line is 1 (the minimal value), and ranges up to 2 for a plane-filling curve or Brownian motion.

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