Reconstructing a Rotor from Initial and Final Frames using
Characteristic Multivectors: with applications in Orthogonal
Transformations: Version 2
Abstract
If an initial frame of vectors
$\{e_i\}$ is related to a final frame
of vectors $\{f_i\}$ by, in geometric
algebra (GA) terms, a {\it rotor}, or in linear algebra
terms, an {\it orthogonal transformation}, we often
want to find this rotor given the initial and final sets of vectors. One
very common example is finding a rotor or $4\times 4$
orthogonal matrix representing rotation and translation, given knowledge
of initial and transformed points. In this paper we discuss methods in
the literature for recovering such rotors and then outline a GA method
which generalises to cases of any signature and any dimension, and which
is not restricted to orthonormal sets of vectors. The proof of this
technique is both concise and elegant and uses the concept of
{\it characteristic multivectors} as discussed in the
book by Hestenes \& Sobczyk, which contains a treatment
of linear algebra using geometric algebra. Expressing orthogonal
transformations as rotors, enables us to create {\it
fractional transformations} and we discuss this for some classic
transforms. In real applications, our initial and/or final sets of
vectors will be noisy. We show how to use the characteristic multivector
method to find a ‘best fit’ rotor between these sets and compare our
results with other methods.