loading page

Reconstructing a Rotor from Initial and Final Frames using Characteristic Multivectors: with applications in Orthogonal Transformations: Version 2
  • Anthony Lasenby,
  • Joan Lasenby,
  • Charalampos Matsantonis
Anthony Lasenby
University of Cambridge

Corresponding Author:a.n.lasenby@mrao.cam.ac.uk

Author Profile
Joan Lasenby
Cambridge University
Author Profile
Charalampos Matsantonis
University of Cambridge
Author Profile


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.
30 Jan 2022Submitted to Mathematical Methods in the Applied Sciences
01 Feb 2022Submission Checks Completed
01 Feb 2022Assigned to Editor
07 Feb 2022Reviewer(s) Assigned
03 Jun 2022Review(s) Completed, Editorial Evaluation Pending
04 Jun 2022Editorial Decision: Revise Major
28 Aug 20221st Revision Received
29 Aug 2022Submission Checks Completed
29 Aug 2022Assigned to Editor
29 Aug 2022Reviewer(s) Assigned
13 Sep 2022Review(s) Completed, Editorial Evaluation Pending
13 Sep 2022Editorial Decision: Accept