# Sinuosity

Calculation of sinuosity for an oscillating curve.

Sinuosity or sinuosity index or sinuosity coefficient of a continuously derivable curve having at least one inflection point, is the ratio of the curvilinear length (along the curve) and the distance (straight line) between the end points of the curve:

$\frac{{\text{actual path length}}}{{\text{shortest path length}}}$

This sinuosity index is 1 in the case in which the actual path length is equal to the shortest path length, and can increase to infinity for a closed loop (where the shortest path length is zero) or for an infinitely-long actual path.[1]

The sinuosity of a sine function (over a whole number of half-periods) can be calculated to be 1.216.

## Rivers

The meandering Rio Cauto at Guamo Embarcadero, Cuba, is not taking the shortest path downslope. Therefore, its sinuosity, or sinuosity index, is > 1.

In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by:

$\text{SI} = \frac{{\text{channel length}}}{{\text{downvalley length}}}$

The difference from the general form happens because the downvalley path is not perfectly straight. The sinuosity index can be explained, then, as the deviations from a path defined by the direction of maximum downslope. For this reason, bedrock streams that flow directly downslope have a sinuosity index of 1, and meandering streams have a sinuosity index that is greater than 1.[2]

## Remarkable values

Example with 270° angle

With similar opposite arcs joints in the same plane, continuously differentiable:

Central angle Sinuosity
30° $\frac{\pi}{6}$ $\frac{\pi}{3(\sqrt{6}-\sqrt{2})}$ 1.0115
60° $\frac{\pi}{3}$ $\frac{\pi}{3}$ 1.0472
90° $\frac{\pi}{2}$ $\frac{\pi}{2\sqrt{2}}$ 1.1107
120° $\frac{2\cdot\pi}{3}$ $\frac{2\cdot\pi}{3\sqrt{3}}$ 1.2092
150° $\frac{5\cdot\pi}{6}$ $\frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})}$ 1.3552
180° $\pi$ $\frac{\pi}{2}$ 1.5708
210° $\frac{7\cdot\pi}{6}$ $\frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})}$ 1.8972
240° $\frac{4\cdot\pi}{3}$ $\frac{4\cdot\pi}{3\sqrt{3}}$ 2.4184
270° $\frac{3\cdot\pi}{2}$ $\frac{3\cdot\pi}{2\sqrt{2}}$ 3.3322
300° $\frac{5\cdot\pi}{3}$ $\frac{5\cdot\pi}{3}$ 5.2360
330° $\frac{11\cdot\pi}{6}$ $\frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})}$ 11.1267

## References

1. ^ Leopold, Luna B., Wolman, M.G., and Miller, J.P., 1964, Fluvial Processes in Geomorphology, San Francisco, W.H. Freeman and Co., 522p.
2. ^ Mueller, Jerry (1968). "An Introduction to the Hydraulic and Topographic Sinuosity Indexes1". Annals of the Association of American Geographers 58 (2): 371. doi:10.1111/j.1467-8306.1968.tb00650.x.