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[parent] decomposition of orthogonal operators as rotations and reflections (Theorem)
DecompositionOfOrthogonalOperatorsAsRotationsAndReflections

"decomposition of orthogonal operators as rotations and reflections" is owned by stevecheng.
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See Also: rotation matrix, orthogonal matrices, dimension of the special orthogonal group, Rodrigues' rotation formula, derivation of rotation matrix using polar coordinates, derivation of 2D reflection matrix

Keywords:  rotation, reflection, orthogonal

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Cross-references: reflects, angle of rotation, coordinate vector, degree of freedom, coordinates, right, embedding, Rodrigues rotation formula, relation, obvious, composition, axis, determinant, positive, language, strictly, act on, component, number, parity, contains, decomposition, formula, normal, unitary, structure, inner product, orthonormal, rotation matrix, unit vector, fix, even, complete, induction hypothesis, plane, line, restricted, orthogonal complement, identity, induction, factor, linear combinations, sinusoid, exponential, solutions, equations, disjoint, differential equation, consequence, simple, matrix, coefficient, linear differential equation, theorem, linearly independent, components, imaginary, eigenvalue, eigenvector, fundamental theorem of algebra, invariant, linear operator, elements, complexification, vector space, transformation, point, eigenvectors, dimensions, eigenvalues, vectors, basis, orientation, information, conjugate, occur in, complex roots, complex plane, multiplication, roots, complex, characteristic polynomial, representation, orthonormal basis, angle, proof, subspaces, reflections, rotations, series, operator, orthogonal, inner product space, real
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This is version 12 of decomposition of orthogonal operators as rotations and reflections, born on 2005-07-19, modified 2006-06-16.
Object id is 7242, canonical name is DecompositionOfOrthogonalOperatorsAsRotationsAndReflections.
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AMS MSC15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

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Rotations in n-dimensional space by fourdplus on 2007-07-24 23:33:45
RE: decomposition of orthogonal operators as rotations and reflections
BY: stevecheng.


The article by stevecheng answers a question that I have encountered: Given a rigid n-dimensional body in an arbitrary orientation, how to transform it to another arbitrary orientation by rotations. The answer is: Find the matrix that will do the job (I know the start and end orientations, so that is straightforward), then decompose that matrix to find the rotation planes and the angles of rotation within those planes. Stevecheng shows how.


Here is a supplementary question, I would be grateful if someone could tell me the answer. If we do the above calculation and obtain a set of rotation angles, we can add the absolute values of the angles and arrive at a total angle of rotation. Is this total angle of rotation the minimum possible to achieve the transformation? Obviously there are an infinite number of ways the transformation could be achieved, including using different planes of rotation. It seems intuitive to me that the method of the article gives the best solution, but is this so?

If the answer is yes, I can use the total angle of rotation as a measure of the 'distance' between two arbitrary orientations.

Thanks in advance for any contribution.
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