| 1. | As it odd ), or another kind of improper rotation.
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| 2. | When " 1, the matrix is an improper rotation.
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| 3. | For improper rotations, the points are flipped about the origin.
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| 4. | Then, any orthogonal matrix is either a rotation or an improper rotation.
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| 5. | An improper rotation of an object thus produces a rotation of its mirror image.
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| 6. | The orthogonal group, consisting of all proper and improper rotations, is generated by reflections.
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| 7. | In a wider sense, an improper rotation may be defined as any "'glide plane.
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| 8. | For even it preserves orientation, while for odd it is reversed ( see also improper rotation ).
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| 9. | Like " D nd " it contains a number of improper rotations without containing the corresponding rotations.
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| 10. | This decomposition is not, however, stable under improper rotations ( e . g ., reflections in a hyperplane ).
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