I'm trying to understand what the signs of a 2d translation matrix would be when you flip the y axis upside down. I have a camera system whose axis are defined with y pointing in the direction of gravity instead of up. It comes with a set of three euler angles in this coordinate system and I'm trying to learn how to convert them into a 3x3 matrix. But for now I'm looking at the 2D case just to understand.
So I tried to learn how this works for the normal case.
Here my gray vector R is defined like this:
$R = \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}cosθ \\sinθ\end{bmatrix}$
My yellow vector is then
$R' = \begin{bmatrix}cos(θ+ω) \\sin(θ+ω)\end{bmatrix}$
I can expand that with trigonometric identity to be:
$R' = \begin{bmatrix}cos(θ)cos(ω) -sin(θ)sin(ω)\\ \\sin(θ)cos(ω) + cos(θ)sin()\end{bmatrix}$
then I can substitute cos(θ) for x and sin(θ) for y and get.
$R' = \begin{bmatrix}xcos(ω) -ysin(ω)\\ \\ycos(ω) + xsin()\end{bmatrix}$
Finally this can also be written as:
$R' = \begin{bmatrix} x \\ y \end{bmatrix}\begin{bmatrix}cos(ω) -sin(ω)\\ \\sin(ω) + cos(ω)\end{bmatrix}$
and that's my 2D rotation matrix for the normal case.
So then what if I have the y axis upside down like this? Immediately I'm not sure if I've drawn my vectors in the right quadrant or if I'm using the right rotation direction (clockwise now instead of counter clockwise before).
Assuming that is okay this time I get.
$R = \begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}cosθ \\-sinθ\end{bmatrix}$
My yellow vector is then
$R' = \begin{bmatrix}cos(θ+ω) \\-sin(θ+ω)\end{bmatrix}$
I can expand that with trigonometric identity to be:
$R' = \begin{bmatrix}cos(θ)cos(ω) -sin(θ)sin(ω)\\ \\-sin(θ)cos(ω) - cos(θ)sin(ω)\end{bmatrix}$
which simplifies to
$R' = \begin{bmatrix}xcos(ω) -ysin(ω)\\ \\-ycos(ω) - xsin(ω)\end{bmatrix}$
and
$R' = \begin{bmatrix}x\\y\end{bmatrix}\begin{bmatrix}cos(ω) -sin(ω)\\ \\ - sin(ω)-cos(ω)\end{bmatrix}$
I'm really not sure if this is the right thing do or not, or perhaps I've screwed up my rotation direction or my signs somewhere.
