Plotting ellipses with parametric curves

Hi all!

I need to plot the following ellipses on the same plot:

Ellips 1:

Ellipse 2:

I have substituted the formula for the second ellipse by u and v:

u =

v =

Ellipse 2:

I have used the following parametrisation:

For an ellipse with the general fomula

the parametrization: and

So, I have rearranged the equation of both ellipses to fit the general formula and then calculated a and b according the get the parametrization above. I was supposed to get the following plot:

Instead I have got this:

I have written the following code:

% First ellipse

t = linspace(0,2*pi,200);
a = sqrt(2);
b = sqrt(2/3);

x = a.*cos(t);
y = b.*sin(t);

plot(((1/2)*(x.^2)), ((3/2).*(y.^2)), '-k', 'LineWidth', 1.5)
axis equal
hold on

% Second ellipse
t = linspace(0,2*pi,200);
a = 2;
b = 1;

x = (-a/3).*(cos(t)+((-2*b/3).*sin(t)));
y = (((2*a/3).*cos(t))+((7*b/3).*sin(t)));
u = x-(2.*y);
v = (2.*x)+y;

plot(((1/4).*(u.^2)), ((v.^2)), '-k', 'LineWidth', 1.5)
axis equal
hold off

What is it that I'm doing wrong? I am very thankful for all the help in advance!

Answer 1

[Chunru]

% First ellipse
t = linspace(0,2*pi,200);
a = sqrt(2);
b = sqrt(2/3);
x = a.*cos(t);
y = b.*sin(t);
%plot(((1/2)*(x.^2)), ((3/2).*(y.^2)), '-k', 'LineWidth', 1.5)
plot(x, y, '-k', 'LineWidth', 1.5)
axis equal
hold on
% Second ellipse
t = linspace(0,2*pi,200);
a = 2;
b = 1;
u = a.*cos(t);
v = b.*sin(t);
xy= ([1 -2; 2 1])\[u; v];    % since [u; v] = [1 -2; 2 1][x; y]

xy = 2×200

0.4000 0.4124 0.4244 0.4360 0.4472 0.4579 0.4682 0.4780 0.4873 0.4961 0.5044 0.5123 0.5196 0.5264 0.5327 0.5384 0.5436 0.5483 0.5524 0.5560 0.5590 0.5615 0.5634 0.5647 0.5655 0.5657 0.5653 0.5644 0.5629 0.5609 -0.8000 -0.7933 -0.7858 -0.7775 -0.7684 -0.7586 -0.7480 -0.7367 -0.7246 -0.7118 -0.6984 -0.6842 -0.6693 -0.6537 -0.6376 -0.6207 -0.6033 -0.5852 -0.5666 -0.5474 -0.5277 -0.5074 -0.4866 -0.4654 -0.4436 -0.4215 -0.3989 -0.3759 -0.3525 -0.3288

x = xy(1, :);
y = xy(2, :);
%plot(((1/4).*(u.^2)), ((v.^2)), '-g', 'LineWidth', 1.5)
plot(x, y, '-g', 'LineWidth', 1.5)
axis equal
hold off

[Szabolcs Simon-Guth]

Thank you very much for the help! Apperantly, I have not done the variable transformation matrix. Thank you again! :)