The Lemniscate of Bernoulli

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 x y k x y 2 2 2 2 2

The Parameterization of the “Lemniscate of Bernoulli”

x y a x y 22 2 2 2 2

c, sin,osx y r x r y r 2 2 2

cos sinr r 22 2 2 2

Cartesian equation:

cos sinr 2 2 2

cosr 2 2

cosr 2

cos cosx 2

sin cosy 2

We have,

Thus, the parametric equations are:

a 1

Using the equations of transformation...

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);

The Area of the Lemniscate of Bernoulli

Polar equation:

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, 2a units apart, is constant and

equal toa2

[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)

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)

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.

x a x y 4 2 2 2:Cartesian Equation

The Lemniscate of Gerono: Parameterization

cos , sin ,x r y r a 1

x x y4 2 2

cos cos sinr r r 4 4 2 2 2 2

cos cos sinr r 4 4 2 2 2

cos cosr r 4 4 2 2

cos cosr 2 4 2

sec cosr 4 2Thus, the Parametric equations are,

sec cos cos

sec cos sin

x

y

4

4

2

2

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');

2

34

4

cosr a 2 2 2

0

0

Lemniscate of Gerono

Polar Curve

a 1

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

The “Lemniscate of Booth”

ycx y y x 22 2 2 2 24 4

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”

c 1 c 1c 0 1

.c5

[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);