2007/01/26 by Johan Ernest Mebius, Mebius, Johan Ernest · 1 citation
Mathematics · #20G20 (Primary) #51N20 (Secondary) #FOS: Mathematics #General Mathematics (math.GM) #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.math/0701759
openalex publication_date 2007/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised correspondingly. Inequalities involving the angles through which the coordinate axes in 3D space are displaced are used to prove that the left-hand and the right-hand quaternions are each other's inverses, thus proving the Euler-Rodrigues formula. A general procedure to determine the Euler parameters of a given 3D rotation matrix is sketched. By equating the leftmost top element to -1 instead of +1 in the general 4D rotation matrix, one proves the counterpart of the Euler-Rodrigues formula for 3D rotoreflections. Keywords: Euler--Rodrigues formula, Euler parameters, quaternions, four--dimensional rotations, three--dimensional rotations, rotoreflections