How to apply rotation to an ellipse defined by center and axis lengths?

Question

This is embarassing, but however. My ellipse is defined by:

centerX and centerY;
majorAxis and minorAxis, lengths of both axis;
orientation, angle between the horizontal X axis and the major axis of the ellipse.

I started plotting the ellipse with its major axis parallel to the X axis:

theta = 0 : 0.05 : 2*pi;
orientation=orientation*pi/180;
xx = (majorAxis/2) * sin(theta) + centerX;
yy = (minorAxis/2) * cos(theta) + centerY;
%plot(xx,yy)

Now I only have to rotate this stuff by orientation degrees or radiants. Any help? I tried a lot of stuff, e.g.:

plotting and then rotateing the object;
defining a rotation matrix R = [cos(orientation), -sin(orientation); sin(orientation), cos(orientation)]; and applying it to xx-centerX and yy-centerY.

For some reason I can't get desired results. Of course I need points to be rotated around the center, not the origin.

Side note: if this is any help, I'm trying to plot ellipses defined by regionprops, using the Centroid,MajorAxisLength,MinorAxisLength and Orientation properties.

To @Luis Mendo, here's what I get:

enter image description here

and this is another try:

enter image description here

For the bottom feature I'm getting orientation = 8 degrees, more or less, which is totally reasonable. But using your code, which I'm posting below, I get a -8 ellipse!

       theta = 0 : 0.05 : 2*pi;
       orientation=orientation*pi/180;
       delta_x = (majorAxis/2) * sin(theta);
       delta_y = (minorAxis/2) * cos(theta);
       delta_x_ok = delta_x*cos(orientation) - delta_y*sin(orientation);
       delta_y_ok = delta_x*sin(orientation) + delta_y*cos(orientation);
       plot(delta_x_ok + centerX, delta_y_ok + centerY, 'r', 'LineWidth', 1.3);

Issue with the y-axis was tricky, but can't believe this got upvoted ;) — natario

Luis Mendo Luis Mendo · Accepted Answer · 2015-03-31T11:38:14

xx-centerX and yy-centerY can be interpreted as coordinates with respect to axes aligned and centered with the rotated ellipse. To obtain coordinates with respect to the actual (non-rotated, non-centered) axes, say xx2 and yy2, you only need to apply the transformation

xx2 = (xx-centerX)*cos(orientation) - (yy-centerY)*sin(orientation) + centerX;
yy2 = (xx-centerX)*sin(orientation) + (yy-centerY)*cos(orientation) + centerY;

Then plot the rotated ellipse with

plot(xx2,yy2)

Example:

majorAxis = 2;
minorAxis = 1;
centerX = 10;
centerY = 15;
orientation = -45;

theta = 0 : 0.05 : 2*pi;
orientation=orientation*pi/180;
xx = (majorAxis/2) * sin(theta) + centerX;
yy = (minorAxis/2) * cos(theta) + centerY;

xx2 = (xx-centerX)*cos(orientation) - (yy-centerY)*sin(orientation) + centerX;
yy2 = (xx-centerX)*sin(orientation) + (yy-centerY)*cos(orientation) + centerY;
plot(xx2,yy2)
axis equal
grid

enter image description here

The gap in the ellipse is produced because the theta step does not divide 2*pi. To correct that, use for example

theta = linspace(0, 2*pi, 150);

which gives

enter image description here

How to apply rotation to an ellipse defined by center and axis lengths?

2 Answers