Rotation Transformation

#Math

$$

\text{rot}(\vec x)=\begin{bmatrix} \cos\theta&-\sin\theta\ \sin\theta&\cos\theta \end{bmatrix}\vec x= \begin{bmatrix} a&-b\ b&a \end{bmatrix}\vec x,\space a^2+b^2=r^2
$$

• $\theta$ is the rotation from the x-axis toward the y-axis
• $r$ is the scaling factor of the vector, with 1 being no scaling
• One can imagine the transformation as scaling and rotating the vector$\begin{bmatrix} 1\ 0 \end{bmatrix}$to$\begin{bmatrix} a\ b \end{bmatrix}$, transforming the curve/coordinate plane together as well