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On the Rigid Rotation Concept in the n-Dimensional Spaces
by
Daniele Mortari
Texas A&M University, College Station, TX
A general mathematical formulation of the n×n proper orthogonal matrix, that corresponds to a simple rigid rotation in n-Dimensional space, is here given. It is shown that a simple rigid rotation depends on an angle (principal angle) and on a set of (n-2) principal axes. The latter, however, can be more conveniently replaced by only 2 orthogonal directions that identify the plane of rotation. The inverse problem, that is, how to compute these principal rotation parameters from the rotation matrix, is also treated. Then, the Euler Theorem is fully extended to rotations in n-Dimensional spaces by a constructive proof that establishes the elegant relationship between orientation of the displaced orthogonal axes and a minimum set of simple rigid rotations. This fundamental relationship, which introduces a new decomposition for proper orthogonal matrices (those identifying an orientation), can be canonically expressed by either a product or a sum of the same simple rotation matrix set. The parallel decomposition, associated with the skew-symmetric matrices, is also given. Finally, the ortho-skew matrices, which are at once orthogonal and skew-symmetric, are introduced. These ortho-skew matrices exist in the even dimensional spaces only; they are shown to represent the extension of the imaginary unit to the matrix field.
Date received: December 14, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadd-04.