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Extension of Euler's theorem to n-dimensional spacesEuler's theorem states that any sequence of finite rotations of a rigid body can be described as a single rotation of the body about a fixed axis in three-dimensional Euclidean space. The usual statement of the theorem in the literature cannot be extended to Euclidean spaces of other dimensions. Equivalent formulations of the theorem are given and proved in a way which does not limit them to the three-dimensional Euclidean space. Thus, the equivalent theorems hold in other dimensions. The proof of one formulation presents an algorithm which shows how to compute an angular-difference matrix that represents a single rotation which is equivalent to the sequence of rotations that have generated the final n-D orientation. This algorithm results also in a constant angular velocity which, when applied to the initial orientation, eventually yields the final orientation regardless of what angular velocity generated the latter. The extension of the theorem is demonstrated in a four-dimensional numerical example.
Document ID
19900035444
Acquisition Source
Legacy CDMS
Document Type
Reprint (Version printed in journal)
External Source(s)
Authors
Bar-Itzhack, Itzhack Y.
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Date Acquired
August 14, 2013
Publication Date
November 1, 1989
Publication Information
Publication: IEEE Transactions on Aerospace and Electronic Systems
Volume: 25
ISSN: 0018-9251
Subject Category
Numerical Analysis
Accession Number
90A22499
Distribution Limits
Public
Copyright
Other

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