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I have a rectangle - lets say 100 long by 75 high. Origin been bottom left corner. I move the rectangle up and across by 10 and rotate by 3 degrees from centre of part. How do I calculate the new position of the origin in X and Y.

I need to make only 3 measurements - 2 from X direction - 1 from Y. I will use a laser to get 3 master measurements. I will move part and re-measure from same positions. I can compare new measurements to original but do not know to calculate.

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It is easier if you make the origin be the center of the rectangle. Then a point that starts at $(a,b)$ will move by the translation to $(a-10,b+10)$ A rotation of $3^\circ$ will move that to $((a-10)\cos 3^\circ-(b+10) \sin 3^\circ,(a-10)\sin 3^\circ + (b+10) \cos 3^\circ)$

It is not clear what you are asking in the second part. It seems disconnected from the first.

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