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Lemniscates. Lemniscates. The Lemniscate of Bernoulli.

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slide1

Lemniscates

Lemniscates

slide2

The

Lemniscate

of

Bernoulli

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Jacob Bernoulli first described his curve in 1694 as a modification of an ellipse. He named it the “Lemniscus", from the Latin word for “pendant ribbon”, for, as he said, it was “Like a lying eight-like figure, folded in a knot of a bundle, or of a lemniscus, a knot of a French ribbon”. At the time he was unaware of the fact that the lemniscate is a special case of the “Cassinian Oval”, described by Cassini in 1680. The original form that Bernoulli studied was the locus of points satisfying the equation

slide4

The Parameterization of the “Lemniscate of Bernoulli”

Cartesian equation:

Using the equations of transformation...

We have,

Thus, the parametric equations are:

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theta = 0:.005:2*pi ;

x = cos(theta).*sqrt(cos(2.*theta));

y = sin(theta).*sqrt(cos(2.*theta));

h = plot(x,y); axis equal

set(h,\'Color\',‘r‘,\'Linewidth\',3);

xl = xlabel(\'0 \leq \theta \leq 2\pi\',\'Color\',‘k\');

set(xl,\'Fontname\',\'Euclid\',\'Fontsize\',18);

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The Lemniscate of Bernoulliis a special case of the “Cassinian Oval”, which is the locus of a point P, the product of whose distances from two focii, 2a units apart, is constant and equal to

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[x,y] = meshgrid(-2*pi:.01:2*pi);

a = 5;

z = sqrt((x-a).^2+y.^2).*sqrt((x+a).^2+y.^2);

contour(x,y,z,25); axis(\'equal’,’square’);

xl = xlabel(\'-2\pi \leq {\it{x,y}} \leq 2\pi\');

set(xl,\'Fontname\',\'Euclid\',\'Fontsize\',14);

title(\'The Cassinian Oval\',\'Fontsize\',12)

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a = 2; b = 2;

[x,y] = meshgrid(-5:.01:5);

colormap(\'jet\');axis equal

z = ((x-a).^2+y.^2).*((x+a).^2+y.^2)-b^4;

contour(x,y,z,0:6:60);

set(gca,\'xtick\',[],\'ytick\',[]);

xl = xlabel(\'-2\pi \leq {\it{x,y}} \leq 2\pi\');

set(xl,\'Fontname\',\'Euclid\',\'Fontsize\',14);

title(\'The Cassinian Oval\'Fontsize\',12)

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The “Lemniscate of Gerono” is named for the French mathematician Camille – Christophe Gerono (1799 – 1891). Though it was not discovered by Gerono, he studied it extensively. The name was officially given in 1895 by Aubry.

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The Lemniscate of Gerono: Parameterization

Thus, the Parametric equations are,

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theta = 0:.001:2*pi ;

r = (sec(theta).^4.*cos(2.* theta)).^(1/2);

x = r.*cos(theta);

y = r.*sin(theta);

plot(x,y,\'color\',[.782 .12 .22],\'Linewidth\',3);

set(gca,\'Fontsize\',10);

xl = xlabel(\'0 \leq \theta \leq 2\pi\');

set(xl,\'Fontname\',\'Euclid\',\'Fontsize\',18,\'Color\',\'k\');

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Construction of the Lemniscate of Gerono

Let there be a unit circle centered on the origin. LetPbe a point on the circle. Let Mbe the intersection of x = 1and a horizontal line passing throughP.Let Qbe the intersection of the line OMand a vertical line passing through P. The trace of Q as Pmoves around the circle is the Lemniscate of Gerono.

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The “Lemniscate of Booth”

When the curve consists of a single oval, but when

it reduces to two tangent circles. When the curve becomes a lemniscate, with the case ofproducing the “Lemniscate of Bernoulli”

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[x,y] = meshgrid(-pi:.01:pi);

c = (1/4)*((x.^2+y.^2)+(4.*y.^2./(x.^2+y.^2)));

contour(x,y,c,12); axis(‘equal’,’square’);

set(gca,\'xtick\',[],\'ytick\',[]);

xl = xlabel(\'-\pi \leq {\it{x,y}} \leq \pi\');

set(xl,\'Fontname\',\'Euclid\',\'Fontsize\',9);

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