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RIGHT LINTEARIA
Curve studied by Euler
in 1744, Sturm
in 1840.
Other names: rectangular Sturm roulette, harmonic elastic curve. 
The right lintearia is the lintearia (or elastic
curve) the inflexion points of which are orthogonal to the translation
axis.
Differential equation:
i.e.
Cartesian parametrization: With , we get the complete curve as , where q is an integer. Curvilinear abscissa: ; radius of curvature: . 

It is the solution of many geometrical or physical problems,
two examples of which follow:
The right lintearia is the locus of the centre of a rectangular hyperbola rolling without slipping on a line, hence the name of rectangular Sturm roulette. 

It is also the solution to the following calculus of
variation problem: finding a curve with given length L joining two
(variable) points A and B on a (fixed) line (D), for
which the rotation around the line (D) generates the solid with
maximal volume.
Note that the curve with given length L joining two (variable) points A and B on a (fixed) line (D), delimiting the maximal area, is a semicircle. This property could explain the shape of the cornea (see this article), and that of an inflatable Mylar balloon (see this article.). 

Below, a comparison of the right lintearia, the semicircle,
a square, and a rectangle, joining A to B.
We write b = AB/2 and a the distance
of the furthermost point on the line joining A and B to (AB).
The area delimited by the curve is noted S, and
the volume of the solid of revolution V.
Right lintearia  Semicircle  Double square rectangle  Square 




b = a I with
L = a J with 
b = a

b = a
L = 4 a

b = a/2
L = 3 a

The maximal volume is obtained with the lintearia, but
among the rectangles, the maximum is obtained with the square.
The maximal area is obtained with the semicircle, but
among the rectangles, the maximum is obtained with the double square.
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© Robert FERRÉOL 2017