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    <p>So, after investigating many claimed solutions, I have finally found a method that works; The answer by <a href="https://stackoverflow.com/users/671973/andri">Andri</a> and <a href="https://stackoverflow.com/users/2971/magnus-hoff">Magnus Hoff</a> on <a href="https://stackoverflow.com/questions/5789239/calculate-largest-rectangle-in-a-rotated-rectangle#7519376">Calculate largest rectangle in a rotated rectangle</a>.</p> <p>The below Python code contains the method of interest - <code>largest_rotated_rect</code> - and a short demo.</p> <pre><code>import math import cv2 import numpy as np def rotate_image(image, angle): """ Rotates an OpenCV 2 / NumPy image about it's centre by the given angle (in degrees). The returned image will be large enough to hold the entire new image, with a black background """ # Get the image size # No that's not an error - NumPy stores image matricies backwards image_size = (image.shape[1], image.shape[0]) image_center = tuple(np.array(image_size) / 2) # Convert the OpenCV 3x2 rotation matrix to 3x3 rot_mat = np.vstack( [cv2.getRotationMatrix2D(image_center, angle, 1.0), [0, 0, 1]] ) rot_mat_notranslate = np.matrix(rot_mat[0:2, 0:2]) # Shorthand for below calcs image_w2 = image_size[0] * 0.5 image_h2 = image_size[1] * 0.5 # Obtain the rotated coordinates of the image corners rotated_coords = [ (np.array([-image_w2, image_h2]) * rot_mat_notranslate).A[0], (np.array([ image_w2, image_h2]) * rot_mat_notranslate).A[0], (np.array([-image_w2, -image_h2]) * rot_mat_notranslate).A[0], (np.array([ image_w2, -image_h2]) * rot_mat_notranslate).A[0] ] # Find the size of the new image x_coords = [pt[0] for pt in rotated_coords] x_pos = [x for x in x_coords if x &gt; 0] x_neg = [x for x in x_coords if x &lt; 0] y_coords = [pt[1] for pt in rotated_coords] y_pos = [y for y in y_coords if y &gt; 0] y_neg = [y for y in y_coords if y &lt; 0] right_bound = max(x_pos) left_bound = min(x_neg) top_bound = max(y_pos) bot_bound = min(y_neg) new_w = int(abs(right_bound - left_bound)) new_h = int(abs(top_bound - bot_bound)) # We require a translation matrix to keep the image centred trans_mat = np.matrix([ [1, 0, int(new_w * 0.5 - image_w2)], [0, 1, int(new_h * 0.5 - image_h2)], [0, 0, 1] ]) # Compute the tranform for the combined rotation and translation affine_mat = (np.matrix(trans_mat) * np.matrix(rot_mat))[0:2, :] # Apply the transform result = cv2.warpAffine( image, affine_mat, (new_w, new_h), flags=cv2.INTER_LINEAR ) return result def largest_rotated_rect(w, h, angle): """ Given a rectangle of size wxh that has been rotated by 'angle' (in radians), computes the width and height of the largest possible axis-aligned rectangle within the rotated rectangle. Original JS code by 'Andri' and Magnus Hoff from Stack Overflow Converted to Python by Aaron Snoswell """ quadrant = int(math.floor(angle / (math.pi / 2))) &amp; 3 sign_alpha = angle if ((quadrant &amp; 1) == 0) else math.pi - angle alpha = (sign_alpha % math.pi + math.pi) % math.pi bb_w = w * math.cos(alpha) + h * math.sin(alpha) bb_h = w * math.sin(alpha) + h * math.cos(alpha) gamma = math.atan2(bb_w, bb_w) if (w &lt; h) else math.atan2(bb_w, bb_w) delta = math.pi - alpha - gamma length = h if (w &lt; h) else w d = length * math.cos(alpha) a = d * math.sin(alpha) / math.sin(delta) y = a * math.cos(gamma) x = y * math.tan(gamma) return ( bb_w - 2 * x, bb_h - 2 * y ) def crop_around_center(image, width, height): """ Given a NumPy / OpenCV 2 image, crops it to the given width and height, around it's centre point """ image_size = (image.shape[1], image.shape[0]) image_center = (int(image_size[0] * 0.5), int(image_size[1] * 0.5)) if(width &gt; image_size[0]): width = image_size[0] if(height &gt; image_size[1]): height = image_size[1] x1 = int(image_center[0] - width * 0.5) x2 = int(image_center[0] + width * 0.5) y1 = int(image_center[1] - height * 0.5) y2 = int(image_center[1] + height * 0.5) return image[y1:y2, x1:x2] def demo(): """ Demos the largest_rotated_rect function """ image = cv2.imread("lenna_rectangle.png") image_height, image_width = image.shape[0:2] cv2.imshow("Original Image", image) print "Press [enter] to begin the demo" print "Press [q] or Escape to quit" key = cv2.waitKey(0) if key == ord("q") or key == 27: exit() for i in np.arange(0, 360, 0.5): image_orig = np.copy(image) image_rotated = rotate_image(image, i) image_rotated_cropped = crop_around_center( image_rotated, *largest_rotated_rect( image_width, image_height, math.radians(i) ) ) key = cv2.waitKey(2) if(key == ord("q") or key == 27): exit() cv2.imshow("Original Image", image_orig) cv2.imshow("Rotated Image", image_rotated) cv2.imshow("Cropped Image", image_rotated_cropped) print "Done" if __name__ == "__main__": demo() </code></pre> <p><img src="https://i.stack.imgur.com/76nJM.jpg" alt="Image Rotation Demo"></p> <p>Simply place <a href="https://i.stack.imgur.com/vWdWB.png" rel="noreferrer">this image</a> (cropped to demonstrate that it works with non-square images) in the same directory as the above file, then run it.</p>
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    1. COThe function `largest_rotated_rect` gives rectangle dimensions that cannot be extended, i.e. no axis parallel rectangle bigger in both dimensions will fit into the rotated rectangle. But except for a few special cases this function will not return the largest (maximal area) rectangle dimensions fitting in. See my solution for the true optimum.
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