In either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation.
22.
He wants a conformal bijection for the sphere to the square and two maps f and g are equivalent if f = g? where ? is some rotation or improper rotation of the sphere.
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As the symbol " ? ijk " is a pseudotensor, pseudovector operators are invariant up to a sign : + 1 for proper rotations and " 1 for improper rotations.
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Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation ( 2 reflections ), or an improper rotation ( 3 reflections ).
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However, there is only one irreducible "'pin representation "'( see below ) owing to the non-invariance of the above eigenspace decomposition under improper rotations, and that has dimension 2 k.
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In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation ( or as a sequence of reflections ).
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It is this line that allows an improper rotation to be decomposed into a proper rotation about the line followed by a reflection that flips the line over ( i . e ., in a plane perpendicular to the line ).
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The symmetry group is sometimes also called "'full symmetry group "'in order to emphasize that it includes the orientation-reversing isometries ( like reflections, glide reflections and improper rotations ) under which the figure is invariant.
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While the presence of any symmetry element within a ligand intended for asymmetric induction might appear counterintuitive, asymmetric induction only requires that the ligand be improper rotation axis ); it does not have to be asymmetric ( i . e . devoid of any symmetry element ).
30.
However, one can also consider improper rotations, i . e . a mirror-reflection possibly followed by a proper rotation . ( One example of an improper rotation is inversion through a point in 3-dimensional space . ) Suppose everything in the universe undergoes an improper rotation described by the rotation matrix " R ", so that a position vector "'x "'is transformed to.
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