Sinuosity

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 calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the small intestine), although it is often performed in a plane (with then a possible orthogonal projection of the curve in the selected plan; "classic" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane).

The classification of a sinuosity (e.g. strong / weak) often depends on the cartographic scale of the curve (see the coastline paradox for further details) 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.

The sinuosity S of:

• 2 inverted continuous semicircles located in the same plane is $S\; =\; \backslash tfrac\{\backslash pi\}\{2\}\; \backslash approx\; 1.5708...$. It is independent of the circle radius;
• a sine function (over a whole number n of half-periods), which can be calculated by computing the sine curve's arclength on those periods, is $S\; =\; \backslash textstyle\; \backslash tfrac\{1\}\{n\backslash pi\}\; \backslash int\_\{0\}^\{n\backslash pi\}\; \backslash sqrt\{1\; +\; (\backslash cos\; x)^2\}\; dx\; \backslash approx\; 1.216...$

File:Sinuosite 270.jpg

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

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

: $\backslash text\{SI\}\; =\; \backslash frac\{\{\backslash text\{channel\; length\}\}\}\{\{\backslash text\{downvalley\; length\}\}\}$

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.{{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 = 371–385}}

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

It has been claimed that river shapes are governed by a self-organizing system that causes their average sinuosity (measured in terms of the source-to-mouth distance, not channel length) to be {{pi}},{{citation|title=River Meandering as a Self-Organization Process|last=Stølum|first=Hans-Henrik|journal=Science|year=1996 |volume=271|issue=5256|pages=1710–1713|doi=10.1126/science.271.5256.1710|bibcode=1996Sci...271.1710S|s2cid=19219185 }}. but this has not been borne out by later studies, which found an average value less than 2.{{citation|title=A meandering tale: the truth about pi and rivers|first=James|last=Grime|department=Alex Bellos's Adventures in Numberland|journal=The Guardian|date=March 14, 2015|url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/14/pi-day-2015-pi-rivers-truth-grime}}.

• Curvature
• Oxbow lake

{{reflist}}

{{River morphology}}

Category:Rivers

Category:Curves

Category:Ratios

Category:Curvature (mathematics)