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Lemniscates. Lemniscates. The Lemniscate of Bernoulli.

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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 “ 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 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 ; 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

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 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 of the Lemniscate of Bernoulli

Polar equation:


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The 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 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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[ 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 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 = 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 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 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 “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: 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 Parameterization

Thus, the Parametric equations are,


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theta = 0:.001:2*pi ; 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

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 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

Polar Curve


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Construction of the Lemniscate of Gerono 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

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” 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

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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[ 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 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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