| 1. | The Euclidean group of all rigid motions ( conjugate of the original translation ).
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| 2. | See also subgroups of the Euclidean group.
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| 3. | The set of Euclidean plane isometries forms a composition : the Euclidean group in two dimensions.
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| 4. | The Euclidean group SE ( d ) of direct isometries is generated by translations and rotations.
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| 5. | The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group.
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| 6. | It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise.
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| 7. | These visualizations show the fundamental domains for 2D and 3D Euclidean groups, and 2D spherical groups.
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| 8. | A " rigid body motion " is in effect the same as a curve in the Euclidean group.
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| 9. | Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.
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| 10. | The Euclidean group for SE ( 3 ) is used for the kinematics of a rigid body, in classical mechanics.
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