Drawing a Rotated Rectangle

Question

Hello, I realize this might be more of a math problem.

To draw the lines for my rectangles I need to solve for their corners. I have a rectangle center at (x,y) With a defined Width and Height.

To find the blue points on a non rotated rectangle on top (angle = 0) It is

UL = (x-Width/2),(y+height/2)
UR = (x+Width/2),(y+height/2)
LR = (x+Width/2),(y-height/2)
LL = (x-Width/2),(y-height/2)

How do I find the points if the angle isn't 0?

Thanks in advance.

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Update: although I have (0,0) in my picture as the center point most likely the center point won't be at that location.

Answer 1

If 'theta' is the anti-clockwise angle of rotation, then the rotation matrix is:

| cos(theta)  -sin(theta) |
| sin(theta)   cos(theta) |

i.e.

x' = x.cos(theta) - y.sin(theta)
y' = x.sin(theta) + y.cos(theta)

There's examples of other transformations at http://en.wikipedia.org/wiki/Transformation_matrix

Answer 2

One of the easiest ways to do this is to take the location of the point before rotation and then apply a coordinate transform. Since it's centred on (0,0), this is simply a case of using:

x' = x cos(theta) - y sin(theta)

y' = y cos(theta) + x sin(theta)

Answer 3

See 2D Rotation.

q = initial angle, f = angle of rotation.

x = r cos q 
y = r sin q

x' = r cos ( q + f ) = r cos q cos f - r sin q sin f 
y' = r sin ( q + w ) = r sin q cos f + r cos q sin f

hence:
x' = x cos f - y sin f
y' = y cos f + x sin f

Answer 4

First transform the centre point to 0,0

X' = X-x

Y' = Y-y

Then rotate for an angle of A

X'' = (X-x) * cos A - (Y-y) * sin A

Y'' = (Y-y) * cos A + (X-x) * sin A

Again transform back the centre point to x,y

X''' = (X-x) * cos A - (Y-y) * sin A + x

Y''' = (Y-y) * cos A + (X-x) * sin A + y

Hence compute for all 4 points of (X,Y) with following transformation

X''' = (X-x) * cos A - (Y-y) * sin A + x

Y''' = (Y-y) * cos A + (X-x) * sin A + y

where x, y are the centre points of rectangle and X,Y are the corner points You have n't defined correctly even the corner points when Angle is 0 as I have given in the comments.

After substituting you will get

UL  =  x + ( Width / 2 ) * cos A - ( Height / 2 ) * sin A ,  y + ( Height / 2 ) * cos A  + ( Width / 2 ) * sin A
UR  =  x - ( Width / 2 ) * cos A - ( Height / 2 ) * sin A ,  y + ( Height / 2 ) * cos A  - ( Width / 2 ) * sin A
BL =   x + ( Width / 2 ) * cos A + ( Height / 2 ) * sin A ,  y - ( Height / 2 ) * cos A  + ( Width / 2 ) * sin A
BR  =  x - ( Width / 2 ) * cos A + ( Height / 2 ) * sin A ,  y - ( Height / 2 ) * cos A  - ( Width / 2 ) * sin A

I think this suits your solution.

Answer 5

Rotation matrix (this is becoming a FAQ)