How to map from one coordinate/rotation system to another?

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I'm using an Augmented Reality SDK to define a polygon in 3d space. What the SDK gives me back is a list of poses (3 translation coordinates, and a rotation quaternion), say P0, P1, P2, P3. Each one with a different translation and rotation.

I can save just one of them. Say P0. (the why is irrelevant to my question).

For a different user, the sdk can resolve the saved P0, but with different translation/quaternions because the world origin would be different. Let's call it P0'

I'm struggling to understand how to go about inferring the whole polygon out P0, P1, P2, P3 AND P0'. In other words, I want to calculate the values of P1', P2' and P3'.

I am sorry if the question is naive. I don't even know what keywords to use to Google for hints.