Get unrotated position of a rectangle.

Question

I'm struggeling with the following problem.

I have a rotated rectangular. The following values are known. The size of the rect, the angle the rect is rotated by and the position of point A. The given angle will always be positive.

How can i obtain the unrotated position of point A?

Answer 1

Fix the line between the A and the orange (x) to have length $ L $, and angle $ \theta $ with respect to the horizontal. Let the position of A be the coordinates $ (a_x,a_y) $. Let the other dimension be $ W $.

The center of the rectangle has coordinates $ ( c_x, c_y) = (a_x,a_y) + \frac{L}{2} (-\cos\theta, \sin\theta) + \frac{W}{2} (\sin \theta, \cos \theta) $. Then the position of A after rotation is then $ (a_x',a_y') = (c_x,c_y) + \frac{L}{2} (-1,0) + \frac{W}{2} (0,1) $.