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Can you please help me to figure out a way to obtain global coordinates of a point which is defined within local coordinates system.

(x,y) origin of the local coordinate system.
angle of the local coordinates system relative to global coordinate system.
(x,y) coordinate of the point with in local coordinate system.

Looking for:
(x,y) coordinate of the point with in global coordinate system.

Thank you so much!


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Take $x,y$ the global coordinates, $x^\prime,y^\prime$ the local coordinates, $(h,k)$ the global coordinates of the local system's origin, and $\varphi$ the anticlockwise angle from the global system's horizontal ($x$) to the local system's "horizontal" ($x^\prime$).

What is needed, then, to convert from $(x^\prime,y^\prime)$ to $(x,y)$ is to 1. shift the origin; and 2. rotate by $\varphi$ clockwise. This sequence is most easily expressed in matrix-vector notation:


or explicitly,


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