C++: Rotating a vector around a certain point

Question

I am trying to rotate a vector around a certain point on the vector(in C++):

1 2 3
4 5 6
7 8 9

rotated around the point (1,1) (which is the "5") 90 degrees would result in:

7 4 1
8 5 2
9 6 3

Right now I am using:

x = (x * cos(90)) - (y * sin(90))
y = (y * cos(90)) + (x * sin(90))

But I don't want it rotated around (0,0)

Answer 1

The answer depends on your coordinate system.

Computer graphics coordinate system, with (0,0) at Top left

If you are using a computer graphics vector implementation where (0,0) is the top left corner and you are rotating around the point (dx, dy), then the rotation calculation, including the translation back into the original coordinate system, would be:

x_rotated =      ((x - dx) * cos(angle)) - ((dy - y) * sin(angle)) + dx
y_rotated = dy - ((dy - y) * cos(angle)) + ((x - dx) * sin(angle))

Physics/Maths coordinate system, with (0,0) at Bottom left

If you are using a more traditional real world coordinate system, where (0,0) is the bottom left corner, then the rotation calculation, around the point (dx, dy) including the translation back into the original coordinate system, would be:

x_rotated = ((x - dx) * cos(angle)) - ((y - dy) * sin(angle)) + dx
y_rotated = ((x - dx) * sin(angle)) + ((y - dy) * cos(angle)) + dy

_{Thanks to mmx for their comment on Pesto's post, and to SkeletorFromEterenia for highlighting an error in my implementation.}

Answer 2

The solution is to translate the vector to a coordinate system in which the center of rotation is (0,0). Apply the rotation matrix and translate the vector back to the original coordinate system.

dx = x of rotation center  
dy = y of rotation center

V2 = V - [dx, dy, 0]  
V3 = V2 * rotation matrix  
Result = V3 + [dx, dy, 0]