Coordinates Transformation

Coordinates transformation simply involves the expression of a coordinate, originally expressed in a coordinate system (2), into a new coordinate system (1). When deriving the coordinates transformation, there is the common way which requires to know a cosine equivalence. As a handout, here's, perhaps, a more intuitive way.

From the graph

$\begin{matrix} x_1 & = & \[x_2 - y_2\,\tan\beta]\cos\beta\\ y_1 & = & \[y_2 + x_2\,\tan\beta]\cos\beta \end{matrix}$

but since:

$\tan\beta = \frac{\sin\beta}{\cos\beta}$

we can do:

$\begin{matrix} x_1 & = & x_2\,\cos\beta -y_2\,\sin\beta\\ y_1 & = & x_2\,\sin\beta + y_2 \cos\beta \end{matrix}$

That can be represented in the usual matrix form:

$\begin{bmatrix} x_1\\ y_1 \end{bmatrix} = \begin{bmatrix} x_2\\ y_2 \end{bmatrix}^T\begin{bmatrix} \cos\beta & \sin\beta\\ -\sin\beta & \cos\beta \end{bmatrix}$

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Coordinates Transformation

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