TITLE: How to get coordinates of a point after an image is rotated? (with images)
QUESTION [5 upvotes]: I have a problem that involves a rotating image and finding a previously known point.
Firstly, there is a sequence with the rotation. 


We start with an empty image.
A line is drawn vertically, from (0, 0) to a point in the y-axis (assume 50, possible values range from 0 to 100 which is the max). We will call this point 'a'.
The image is rotated by 'x' degrees (known value).
Another line is drawn vertically, from (0, 0) to a point in the y-axis (assume 60). We will call this point 'b'.


My question is, how do I get the coordinates of point 'a' relative to point (0, 0)?

Thank you all so much and I really appreciate your replies, good or bad. Please do tell me if you need more info on this.

REPLY [2 votes]: Rotating a point $(0,A_y)$ on a plane about the origin by $x$ degrees (counter-clockwise) is given by 
$$
\left[ 
\begin{array}{c}
A_x' \\
A_y' \\ 
\end{array}
\right] 
= 
\left[ 
\begin{array}{cc}
\cos x & -\sin x \\ 
 \sin x & \cos x \\  
\end{array}
\right] 
\left[ 
\begin{array}{c}
0 \\
A_y \\ 
\end{array}
\right],  
$$
where $(A_x',A_y') = (-A_y \sin x, A_y \cos x)$ denotes your new position. Relative to point $B$, the coordinates are:
$$(-A_y \sin x, A_y \cos x - B_y)$$