the lemniscate of bernoulli jacob bernoulli first described his curve in 1694 as a modification of...
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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);