Cartesian coordinate space transformation question

[atsekhan]

I have a question that I am trying to find proof and/or references for:

Suppose we have two sets of points (P1 and P2) in separate N-dimensional Cartesian Spaces S1 and S2.

*** Note: if it can be easily extended to the Euclidean Space - even better.

We need to find Affine Transformation from S1 to S2.

The absolute units of measure are the same in S1 and S2 (that is one of the initial fundamental properties of S1 and S2 I am working with).

My theory is

In order to find the transformation matrix it is sufficient to measure the distances between points D1...Dj that belong to P1 and K1..Kj that belong to P2, such that D1..Dj do not belong to the same N-1 - dimensional plane in S1 and K1..Kj do not belong to the same N-1 dimensional plane in S2.

This is linear algebra, and, while this theory is intuitive, I need a reference or a way to build transformation matrix.

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As an example in 2D space -

suppose we have points A1,A2, A3 that do not lay on the same line (meaning they form a real triangle)

Suppose also we have points B1, B2, B3 and B4 where B1, B2 and B3 do not lay on the same line. The coordinates of B1, B2, B3 and B4 is not the same (meaning the X and Y axis of the B points are not the same as X and Y axis of the A points, and may not necessarily be parallel or orthogonal)

My theory - by knowing the absolute distances between A1..A3 and B1..B3 (all pairs of dots - A1 to B1..B3, A2 to B1..B3 etc) (we will know them from some black-box function that will simply return the absolute distance) we can design a transformation matrix, so coordinates of B4 can be recalculated without measuring the distance between B4 and any other points.

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Would you happen to know whether this is correct, and if yes, provide the reference?

 

[fresh_42]

Cartesian and Euclidean space means the same thing, an $n-$dimensional real vector space. There is nothing said about the points, i.e. there number and relative position to each other, so there cannot be said anything about whether this is possible at all. One can choose one point of each set and consider it as origin of a coordinate system. This reduces the question to find a linear transformation and makes clear, that essential information is missing here.