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ELASTIC CURVE, or ELASTICA, or ELASTIC ROD

right lintearia 
. 0< k < k1 = 0.65222.... 
k = k_{1} = 0.65222....: pseudolemniscate 
k_{1}< k < 1 
k = 1 : convict curve 
k > 1: pseudotrochoid 
Curve studied by Jacques Bernoulli in 1691 who named it elastica, by Euler in 1744, and Poisson in 1833.
Other names: lintearia, radioid with curvature proportional to the abscissa. See on this page of Alain Esculier a program to trace this curve. See also this article on the meanders. 
The elastic curves are the plane curves the curvature of which is, at all points, proportional to the distance to a fixed line, called the directrix.
With the directrix as the y axis, the condition can be written
(1), hence the
Differential equation: . (1) that can be integrated into i.e. . With , we get the parametrization: 
For 1 < k < 1 (figure with k = 1/ 2: tangential angle at the origin equal to  30 °)
If we use the parametrization:

If we use the explicit expression: 
Cartesian parametrization:
With , we get the complete curve as , where q is an integer. 
Here, where is the tangential angle at the origin. Cartesian equation: where ; with , we get the complete curve as , where q is an integer. 
For k = 1:

The integral is no longer elliptic and we have the parametrization:
() which is none other than the convict curve, special case of a syntractrix. 
For k > 1:

Cartesian equation:
where and ; with , the curve can be extended as , where q is an integer. 
According to the YoungLaplace equation, the profile of a rectangular tarpaulin full of water is an elastic curve (the curvature of the tarpaulin being proportional to the distance to the surface of the liquid), hence the name lintearia.
We also find this curve as the trajectory of a torpedo the rudder of which turn with an angle proportional to the depth.
But the elastic curve is also the solution to the following calculus of variations problem: finding, among all the curve with given length, that which minimises the integral of the square of the curvature ; in other words, it is the curve that minimises the variance of the oscillations of the tangent with respect to a fixed direction. It is the reason why it can be found in many natural phenomena.
The meanders of rivers take the shape of elastic curves (see also the meander curve)
Here, the meanders of the Somme river. 
Because of the characteristic property of minimising the curvature, the elastic curve is sometimes used for the layout of roads between two linear portions, hence the name of radioid.
It is also a solution to this other calculus of variations problem: finding a curve of given length joining two given points A and B, for which the rotation around a line (D) coplanar with the line (AB) generates a solid with maximal volume (in other words, maximal for constant).
And it is also characterised by the fact that when a point M describes it at constant speed, the tangent oscillates around M the same way a simple pendulum would oscillate around the parallel to the directrix passing by M (cf. the formula ).
It is the reason why the elastic curve is very similar, for values of k between –1 and 1 to the meander curve (the tangent of which has sinusoidal oscillations), and why they are often mixed up.
See on this German site a very beautiful animation of a generalisation of this curve.
See also the hanging drop of water, generalisation to space of the elastic curve.
The curves such that the curvature is proportional to the distance to a fixed point (instead of a line) are studied on the page of the Norwich spiral; we find, in particular, the lemniscate
of Bernoulli.
The curves such that the curvature is proportional to the curvilinear abscissa are the clothoids.
The curves such that the curvature is proportional to the distance to the intersection point between the normal and a fixed line are studied here.
REMARK: if, in the differential equation , the exponent 3/2 is replaced by 1/2, we get the skipping rope curve.
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© Robert FERRÉOL 2017