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I'd like to recover a rotor in geometric algebra where I know which (multi)vectors transform to given rotated (multi)vectors. I've found many references for 3D/4D dimensions (something like $\sum e_if_i$), but I need a formula for any dimension. Is there an easy one?

Also more importantly, how do I prove that it works?

Because I actually need to rotate multivectors $A_i=ae_i+cf_i$ (or maybe grade-3 basis) to new ones of this form, so that I will need to derive/check that the formula also works for a multivector basis.

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  • $\begingroup$ Are you asking about how to solve the versor equation $X A_i X^{-1} = B_i$ where $X$ is unknown rotor, $A_i$ and $B_i$ are $n$ known multivectors? I think you can solve this equation in a least squares sense by finding the eigenvector corresponding to least eigenvalue of a quadratic form. That works for any dimension. If you don't know how to translate GA multivector expressions to LA matrices check Perwass book: Geometric Algebra With Applications in Engineering $\endgroup$ Jan 23, 2022 at 20:24
  • $\begingroup$ @MauricioCeleLopezBelon This is correct. Thanks. I've posted what I've found so far as an answer. In my particular case I realized in only need low dimensional results, and I guessed the solution for my multivectors. $\endgroup$
    – Gere
    Jan 24, 2022 at 9:18

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For other readers, I'd like to post what I've found out so far:

Apparently, there is an implementation in the Python clifford package of a cartan-daydonny algorithm, which makes me assume there is no closed form solution for higher dimensions.

In the researchgate.net discussion I've found a derivation but there may be an error in the assumptions about rotors making the derivation invalid in higher dimensions. There is another linked paper, but it's beyond what I can understand.

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