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[[Image:Sinuosity.png|thumb|250px|Calculation of '''sinuosity''' for an oscillating curve.]]
[[File:LuzArdidien2003.jpg|thumb|250px|Laces on mountain road with high sinuosity]]


'''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. This [[dimensionless]] quantity is obtained by the following report:


: <math>\frac{{\text{actual path length}}}{{\text{shortest path length}}}</math>
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The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero) or for an infinitely-long actual path.<ref>Leopold, Luna B., Wolman, M.G., and Miller, J.P., 1964, Fluvial Processes in Geomorphology, San Francisco, W.H. Freeman and Co., 522p.</ref>
 
The sinuosity of:
* 2 inverted continuous semicircles located in the same plane is  <math>\pi/2</math> about 1.5708 : it is independent of the circle radius;
* a [[sine]] function (over a whole number of half-periods) can be calculated to be 1.216.
 
The curve must be continuous (no jump) between the two ends. The sinuosity value is really significant when the line is continuously differentiable (no angular point). The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2).
 
The classification of a sinuosity (e.g. strong / weak) often depends on the cartographic scale of the curve and of the object velocity which flowing therethrough (river, avalanche, car, bicycle, bobsleigh, skier, high speed train, etc.): the sinuosity of the same curved line could be considered very strong for a high speed train but low for a river. Nevertheless, it is possible to see a very strong sinuosity in the succession of few river bends, or of laces on some mountain roads.
 
==Rivers==
 
[[Image:rio-cauto-cuba.JPG|thumb|250px|The meandering ''[[Cauto River|Rio Cauto]]'' at Guamo Embarcadero, [[Cuba]], is not taking the shortest path downslope. Therefore, its '''sinuosity index''' is&nbsp;>&nbsp;1.]]
 
In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by:
 
: <math>\text{SI} = \frac{{\text{channel length}}}{{\text{downvalley length}}}</math>
 
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 [[meander]]ing streams have a sinuosity index that is greater than 1.<ref>{{cite journal
| doi = 10.1111/j.1467-8306.1968.tb00650.x
| title = An Introduction to the Hydraulic and Topographic Sinuosity Indexes1
| year = 1968
| author = Mueller, Jerry
| journal = Annals of the Association of American Geographers
| volume = 58
| issue = 2
| pages = 371}}</ref>
 
It is also possible to distinguish the case where the stream flowing on the line could not physically travel the distance between the ends: in some hydraulic studies, this leads to assign a sinuosity value of 1 for a torrent flowing over rocky bedrock along a horizontal rectilinear projection, even if the slope angle varies.
 
For rivers, the conventional classes of sinuosity, SI, are:
*            SI <1.05: almost straight
*    1.05 ≤ SI <1.25: winding
*    1.25 ≤ SI <1.50: twisty
*    1.50 ≤ SI: meandering
 
==Remarkable values==
[[File:Sinuosite 270.jpg|thumb|right|Example with 270° angle]]
With similar opposite arcs joints in the same plane, continuously differentiable:
{| border="1" cellpadding="4" cellspacing="0"
|-----
! style="background:#ffdead;" colspan="2" | [[Central angle]]
! style="background:#ffdead;" colspan="2" | Sinuosity
|-----
| style="background:#efefef;" align="Center" | [[Degree (angle)|Degrees]]
| style="background:#efefef;" align="Center" | [[Radian]]s
| style="background:#efefef;" align="Center" | Exact
| style="background:#efefef;" align="Center" | Decimal
|-----
| align="Center" | 30° || align="Center" | <math>\frac{\pi}{6}</math>
| align="Center" | <math>\frac{\pi}{3(\sqrt{6}-\sqrt{2})}</math> || align="Center" | 1.0115
|-----
| align="Center" | 60° || align="Center" | <math>\frac{\pi}{3}</math>
| align="Center" | <math>\frac{\pi}{3}</math> || align="Center" | 1.0472
|-----
| align="Center" | 90° || align="Center" | <math>\frac{\pi}{2}</math>
| align="Center" | <math>\frac{\pi}{2\sqrt{2}}</math> || align="Center" | 1.1107
|-----
| align="Center" | 120° || align="Center" | <math>\frac{2\cdot\pi}{3}</math>
| align="Center" | <math>\frac{2\cdot\pi}{3\sqrt{3}}</math> || align="Center" | 1.2092
|-----
| align="Center" | 150° || align="Center" | <math>\frac{5\cdot\pi}{6}</math>
| align="Center" | <math>\frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})}</math> || align="Center" | 1.3552
|-----
| align="Center" | 180° || align="Center" | <math>\pi</math>
| align="Center" | <math>\frac{\pi}{2}</math> || align="Center" | 1.5708
|-----
| align="Center" | 210° || align="Center" | <math>\frac{7\cdot\pi}{6}</math>
| align="Center" | <math>\frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})}</math> || align="Center" | 1.8972
|-----
| align="Center" | 240° || align="Center" | <math>\frac{4\cdot\pi}{3}</math>
| align="Center" | <math>\frac{4\cdot\pi}{3\sqrt{3}}</math> || align="Center" | 2.4184
|-----
| align="Center" | 270° || align="Center" | <math>\frac{3\cdot\pi}{2}</math>
| align="Center" | <math>\frac{3\cdot\pi}{2\sqrt{2}}</math> || align="Center" | 3.3322
|-----
| align="Center" | 300° || align="Center" | <math>\frac{5\cdot\pi}{3}</math>
| align="Center" | <math>\frac{5\cdot\pi}{3}</math> || align="Center" | 5.2360
|-----
| align="Center" | 330° || align="Center" | <math>\frac{11\cdot\pi}{6}</math>
| align="Center" | <math>\frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})}</math> || align="Center" | 11.1267
|}
 
==References==
{{reflist}}
 
[[Category:Rivers]]
[[Category:Curves]]
[[Category:Dimensionless numbers]]

Latest revision as of 18:53, 29 March 2014


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