The title might not be too descriptive; I'm not sure how to classify this question. I have a s

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esclaufitbzv

Answered question

2022-02-25

The title might not be too descriptive; I'm not sure how to classify this question.
I have a set of linear equations of the form
ciAi+piBi=E
where Ai,Bi and E are 3 by 1 column vectors and ci and pi are scalars. As the subscripts indicate, I have n such linear equations, where all the scalars and vectors except E change with each one. Another limitation is that all I know for each set of linear equations is Ai and Bi, and I want to find E.
Is there a method for solving these equations?
The equation above is a simplification of the equation
RpTTD=ciAipiRDBi
where RpT is the inverse of a rotation matrix, TD is a translation vector and RD is another rotation matrix. I am interested in finding the rotation matrix Rp. Again, this equation describes the relationship between two homogenious points (Ai and Bi) in 3D space, where the z component is 1 (i.e., the c and p scalars project the points to somewhere in the 3D space other than (x, y, 1)). I have n point-pairs A and B.

Answer & Explanation

Pooja Copeland

Pooja Copeland

Beginner2022-02-26Added 8 answers

This is a homogeneous system of 3n linear equations for the 2n+3 variables ci,pi and the 3 components of E. In general, for n3 the trivial solution (all variables 0) will be the only solution. If n<3, or if the equations happen to be linearly dependent and allow non-trivial solutions, you can find them by subtracting E on both sides and writing out the n 3-component equations as 3n equations. Note that since the system is homogeneous, if there are non-trivial solutions, there's a whole vector space of them.

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