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I need to translate the coordinates of four points in 3d reference frame A (P1, P2, P3, P4) given their distances from four other points in reference frame B (Q1, Q2, Q3, Q4);

So I have distances |P1Q1|, |P2Q2|, |P3Q3|, and |P4Q4| and the xyz coordinates of all P points in the A frame and all Q points in the B frame:

How can I determine the transformation matrix to translate P coordinates to the B reference frame?

This is a similar question as in this thread: Get position of a point with known distance between other points but with four distances between eight different points.

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robeat
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  • Do you know the transformation between the two frames? In this case simply transforms the coordinates of the points $P_i$ from the first to the second frame. The distance between two points does not depend from the coordinate system. – Emilio Novati Sep 19 '16 at 13:23
  • No, I was hoping to solve for the transformation using the known distances between the points in the different frames. – robeat Sep 19 '16 at 13:28
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    What type of transformation are you looking for? – amd Sep 19 '16 at 18:23
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    A general isometry in 3d has 6 real degrees of freedom (e.g. 3 for translation and 3 for rotation), so knowing 4 parameters doesn't uniquely define it. A linear transformation has 9 real degrees of freedom, as can be seen from its 3×3 matrix. You can only hope to find a unique answer if you have at least as many scalar constraints as there are real degrees of freedom. So as amd wrote, what class of transformations are you looking for? Anything with 4 or less rdof? – MvG Sep 19 '16 at 20:33

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