# Coordinate substitution between two cartesian systems

I have multiple objects in my lab which are fully described in two coordinate systems. Both systems are Cartesian and in meters. If they had a common origin but were rotated relative to each other I could easily reconcile the two. Additionally, if it was just a displacement it would be even easier. However the two are offset in rotation and displacement.

What I want to do is find how to convert between the two of them. I would think I would just need the coordinates of the same three objects in the two systems. I know how to do this for two coordinate systems that are rotated OR displaced, but I can't figure out how to do it in a scenario with both rotation and displacement. Anyone got a formula for me or (even better) a link describing how it is done?

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I assume you are rotating with a 2x2 (in 2D for example), well you can just use 3x3s to get translation involved too. It's called homogeneous coordinates. – anon Oct 11 '10 at 19:11
Do you know the translation and rotation, and want to find the coordinates of the points, or vice versa? – Rahul Oct 11 '10 at 19:21
First do a displacement so that the origins coincide; then do a rotation after the displacement. That is, compose a displacement and a rotation. – Arturo Magidin Oct 11 '10 at 19:40