how to draw a ellipse with known its general equation

I want to draw a ellipse with known its general form of equation as follow form:

a*x^2 + b*x*y + c*y^2 +d*x + e*y + f = 0

where a b c d e f are the parameters of above equation.

Here, I want convert the general equation to Parametric equations and then draw it. If you can directly draw it without convert, I also accept it , however, I just do not hope to set the step for x as 1:1:1000 to get another value of y or vice versa. Is anybody can help me, thanks ahead!

[Azzi Abdelmalek]

you said to set x=1:1:1000 this is not possible because x is a solution of your equation

[Walter Roberson]

UTA does not want to set x values and find the corresponding y values: UTA hopes for a parametric solution.

Answer 1

[Walter Roberson]

http://en.wikipedia.org/wiki/Ellipse#Parametric_form_in_canonical_position

Answer 2

[Image Analyst]

Try my demo:

clc;    % Clear the command window.
workspace;  % Make sure the workspace panel is showing.
clearvars;
format longg;
format compact;
fontSize = 20;

% Parameterize the equation.
t = linspace(0, 360,1000);
phaseShift = 20;
xAmplitude = 2;
yAmplitude = 1;
x = xAmplitude * sind(t + phaseShift);
y = yAmplitude * cosd(t);

% Now plot the rotated ellipse.
plot(x, y, 'b-', 'LineWidth', 2);
axis equal
grid on;
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
title('Rotated Ellipses', 'FontSize', fontSize);
text(-1.75, 1.4, 'Parametric --', 'Color', 'b', 'FontSize', fontSize);

% Now plot another ellipse and multiply it by a rotation matrix.
% http://www.maa.org/joma/Volume8/Kalman/General.html
rotationAngle = 30;
transformMatrix = [cosd(rotationAngle), sind(rotationAngle);...
  -sind(rotationAngle), cosd(rotationAngle)]
xAligned = xAmplitude * sind(t);
yAligned = yAmplitude * cosd(t);
xyAligned = [xAligned; yAligned]';
xyRotated = xyAligned * transformMatrix;
xRotated = xyRotated(:, 1);
yRotated = xyRotated(:, 2);
hold on;
plot(xRotated, yRotated, 'g-', 'LineWidth', 2);

% Plot a line at 30 degrees
slope = tand(30);
x1 = min(x(:));
y1 = slope * x1;
x2 = max(x(:));
y2 = slope * x2;
line([x1 x2], [y1 y2], 'Color', 'r');
text(-1.75, 1.25, 'Rotation Matrix --', 'Color', 'g', 'FontSize', fontSize);
text(-1.75, 1.1, '30 Degree Line --', 'Color', 'r', 'FontSize', fontSize);

% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);