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Lemniscates. Lemniscates. The Lemniscate of Bernoulli.

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Lemniscates

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Lemniscates

Lemniscates


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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


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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 Area of the Lemniscate of Bernoulli

Polar equation:


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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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Lemniscate of Gerono

Polar Curve


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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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