# Dictionary Definition

tortuosity n : a tortuous and twisted shape or
position; "they built a tree house in the tortuosities of its
boughs"; "the acrobat performed incredible contortions" [syn:
tortuousness,
torsion, contortion, crookedness]

# User Contributed Dictionary

## English

### Noun

# Extensive Definition

Tortuosity is a property of curve being tortuous
(twisted; having many turns). There have been several attempts to
quantify this property.

## Tortuosity in 2-D

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

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

- \tau = \frac

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 integral of square (or module)
of 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:

- \tau = \frac \cdot \sum\limits_^N

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:

- \frac\log \left( \kappa \right) = \frac

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:

- \tau = \frac

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

In most of these methods digital
filters and approximation by splines can be used to decrease
sensitivity to noise.

## Tortuosity in 3-D

Usually subjective estimation is used. However,
several ways to adapt methods estimating tortuosity in 2-D have
also been tried. The methods include arc-chord ratio, arc-chord
ratio divided by number of inflection
points and integral of square of curvature, divided by length
of the curve (curvature is estimated assuming that small segments
of curve are planar) .

## Applications of tortuosity

Tortuosity of blood
vessels (for example, retinal and cerebral blood
vessels) is known to be used as a medical
sign.

In mathematics, cubic
splines minimize the functional, equivalent to
integral of square of curvature (approximating the curvature as the
second derivative).

In hydrogeology, the
tortuosity refers to the ratio of the diffusivity in the free space
to the diffusivity in the porous
medium (analogous to arc-chord ratio of path).

In acoustics and following
initial works by Maurice
Anthony Biot in 1956, the tortuosity is used to describe
sound
propagation in
fluid-saturated porous media. In such media, when frequency of
the sound wave is high enough, the effect of viscous drag force
between the solid and the fluid can be ignored. In this case,
velocity of sound propagation in the fluid in the pores is non-dispersive
and compared with the value of the velocity of sound in the free
fluid is reduced by a ratio equal to the square root of the
tortuosity. This has been used for a number of applications
including the study of materials for acoustic isolation, and for
oil prospection using acoustics means.

In analytical
chemistry applied to polymers and sometimes small
molecules tortuosity is applied in
Gel Permeation Chromatography (GPC) also known as Size
Exclusion Chromatography (SEC). As with any chromatography it is used
to separate mixtures. In
the case of GPC the separation is based on molecular
size and it works by the use of stationary media with
appropriately dimensioned pores. The separation occurs because
larger molecules take a shorter, less tortuous path and elute more
quickly and smaller molecules can pass into the pores and take a
longer, more tortuous path and elute later.

## References

tortuosity in German: Tortuosität

tortuosity in Spanish: Tortuosidad

tortuosity in Russian:
Извилистость