Does rotation of a rectangle keep it rectangular?

Question

If I rotate a rectangle by 45°, does it stay rectangular or become something else? I mean do 90° angles stay 90°?

I am asking this question because I have some results where the rotated rectangle becomes not so rectangular ... I think I have a problem.

Problem: I think the problem comes from scaling.... I draw the resulting data after rotation in another image, and I have got the rectangle correctly rotated. Maybe it is because I used imagesc to draw the background or axis image in MATLAB...?

Answer 1

I found the solution.

Well, I used to draw the image using a different scaling between Y-axis and X-axis. The solution is to set DataAspectRatio to [1 1 1]. In order to use I have used axis image but after imagesc(image) !

Answer 2

Yes, it stays a rectangle.

Rotations are examples of Euclidean transformations, which preserve angles and distances. A rectangle is defined by angles (consecutive sides are at right angles), so the image of a rectangle is a rectangle.

All shapes are preserved as well, so a rectangle stays a rectangle, a circle stays a circle, an ellipse stays an ellipse, and so on. It even keeps its area.

Answer 3

Yes, it remains a rectangle. The best way to see this would be to draw a rectangle on a piece of paper, push your finger down on any point of that paper, and rotate the paper. The rectangle will still be a rectangle.

Answer 4

If you are not plotting at "true aspect ratio", i.e., if the horizontal and vertical scales of your plot are not identical, a rotation of the plane will not appear on the screen as an isometry; circles will appear to be non-circular ellipses, rectangles will rotated to squashed parallelograms, and so forth.

Here's a rotating square plotted at an aspect ratio of $4:1$:

Answer 5

If rectangles are rigidly rotated all lengths remain same. It looks like a parallelogram, it is due to unequal scaling factor in SF x- and y-.

In a proper rotation/translation, it is just as in a transparent polyester sheet / view graph is displaced on the table and nothing happens relatively between line segments by scaling or distortion.

$ x1 = x \cos t - y \sin t ; \, y1 = SF (x \sin t + y \cos t) $

Aspect Ratio is to be $1$ before and after rotation.