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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 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: We have, Thus, the parametric equations are: Using the equations of transformation...

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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 Bernoulli is a special case of the “Cassinian Oval”, which is the locus of a point P, the product of whose distances from two focii, 2 a 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',[ ],'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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Let there be a unit circle centered on the origin. Let P be a point on the circle. Let M be the intersection of x = 1 and a horizontal line passing through P. Let Q be the intersection of the line OM and a vertical line passing through P. The trace of Q as P moves around the circle is the Lemniscate of Gerono. Construction of 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 of producing 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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