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LINTEARIA

Problem posed and solved by Jacques Bernoulli in 1692.
He named the curve lintearia after the Latin word linteum meaning
linen cloth.
Other named proposed: tarpaulin, by analogy with the catenary. |

The lintearia is the shape taken by the profile of a rectangular tarpaulin attached to two horizontal bars, full of water up to the two bars (two planes limit the flow of water), and placed in a uniform gravitational field. The tarpaulin is supposed to be flexible, infinitely thin, inextensible, and without proper mass. |

This can be simplified to , and, by integrating, we get .

Therefore ,
and, by eliminating the integral, we get the differential equation of the
lintearia: .

With ,
we get the parametrization: ,
for , which is none other
than the elastic curve (swap
*x* and *y*).

Therefore, the lintearia is also the curve for which the curvature is proportional to the depth, result that could have been directly obtained by virtue of the Laplace law.

For 0 < k < 1, the lintearia is open, for k1 = -0.65222.....
< k < 0, it is closed; the value of k1 is the limit value corresponding
to the physical problem.

Remark: for a pressure force still proportional to *y* but also
to the infinitesimal length d*x* instead of d*s*, oriented towards
the bottom, i.e. ,
we would get a sinusoid.

See the dangling drop, generalisation to 3D-space of the lintearia.

Compare with the skipping rope curve.

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© Robert FERRÉOL 2017