tilting an ellipse on a given angle

I am plotting ellipses as attached. My program is working fine but I need to tilt the ellipse on an angle (in degrees) entered by my user. The pivotal point is shown as black dot in attachment.

I am using following code to get each ellipse, few parameters e.g Q,R etc are received from another part of program.

xCenter = 2*Q;

yCenter = 2*R;

xRadius = Q;

yRadius = R;

theta = 0 : 0.01 : 2*pi;

x = xRadius * cos(theta) + xCenter;

y = yRadius * sin(theta) + yCenter;

e=plot(x, y, 'LineWidth', 1);

fill(x,y,'g')

axis square;

xlim([0 2*yCenter]);

ylim([0 2*yCenter]);

Answer 1

[Image Analyst]

You need to introduce a phase shift to get a rotation. See my demo code. It makes a rotated ellipse. Then it uses a second way, a rotation matrix, to rotate that ellipse by a specified angle.

% Creates a rotated ellipse, and then rotates it again by a specified angle using a second method: a rotation matrix.
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.
% https://en.wikipedia.org/wiki/Rotation_matrix
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]);

[Snowleopard]

Thanks for your support, I got to rotate ellipse but I have to rotate them on the pivotal point as showed in attached pic.My ellipses are of different major and minor axis so I am unable to manage that. Earlier I had managed by subtracting yCenter = (2*R)-(R-R1); the major axis differences so it was working, now with each angle the distance is varying so my aim to rotate them relative to pivotal point is not working.

[Image Analyst]

I think maybe you just didn't spend the time to go through the math, so I did it for you:

% Creates a rotated ellipse, and then rotate it again by  specified angles using a second method: a rotation matrix.
clc;    % Clear the command window.
workspace;  % Make sure the workspace panel is showing.
clearvars;
format long g;
format compact;
fontSize = 20;
darkGreen = [0, 0.6, 0];

% Parameterize the equation.
t = linspace(0, 360,1000);
xAmplitude = 1;
yAmplitude = 2.5;
xCenter = 2.5;
yCenter = 5;
xOriginal = xAmplitude * sind(t) + xCenter;
yOriginal = yAmplitude * cosd(t) + yCenter;

% Now plot the rotated ellipse.
plot(xOriginal, yOriginal, 'b-', 'LineWidth', 2);
axis equal
grid on;
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
title('Rotated Ellipses', 'FontSize', fontSize);
xlim([-3, 8]);
ylim([-3, 8]);
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized','OuterPosition',[0 0 1 1]);
drawnow;
hold on;

% Now plot more ellipses and multiply it by a rotation matrix.
% https://en.wikipedia.org/wiki/Rotation_matrix
% For each angle, subtract the center, multiply by the rotation matrix and add back in the center.
for rotationAngle = 10 : 20 : 350
  transformMatrix = [cosd(rotationAngle), sind(rotationAngle);...
    -sind(rotationAngle), cosd(rotationAngle)];
  xAligned = (xOriginal - xCenter);
  yAligned = (yOriginal - yCenter/2);
  xyAligned = [xAligned; yAligned]';
  xyRotated = xyAligned * transformMatrix;
  xRotated = xyRotated(:, 1) + xCenter;
  yRotated = xyRotated(:, 2) + yCenter/2;
  hold on;
  plot(xRotated, yRotated, '-', 'Color', darkGreen, 'LineWidth', 2);
end