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SKIPPING ROPE CURVE

Curve studied by Clebsch in 1860.
Other name: troposkein.
Ref: [greenhill] p. 97.
See also this article.

The skipping rope curve is the shape taken by an inextensible homogeneous flexible massive wire in rotation around an axis attached to two points on this axis (gravity is ignored). It is impossible for the naked eye to tell it and an arch of a sinusoid apart, and yet it is the plot of the Jacobi elliptic function noted sn, and not that of the sine function.

It can be proved that the skipping rope curve with extremities A and B and length L is also the curve of length L joining A to B such that the moment of inertia with respect to the axis (AB) is maximal.
 
Differential equation:  .
Cartesian equation: .
Cartesian parametrization written with the elliptic function of the first kind F (EllipticF in Maple)
with  ;
Cartesian equation written with the Jacobi elliptic function sn (JacobiSN in Maple).
Slope at O
Abscissas of the summits: .
Length of an arch: .

 
With the notation of the opposite figure ( = tension of the wire at M = linear mass density of the wire,  = angular speed of rotation), we write that the sum of the forces at M is zero: .

This simplifies to , which, by integration gives .
Hence  and  i.e.  where .
This second order equation yields the first order equation above.

In the equation, if the exponent 1/2 is replaced by 3/2, we get the lintearia.
For another curve plotting an elliptic function of Jacobi, see wheel-road couple.

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© Robert FERRÉOL 2017