3D Isometries That Leave Origin Fixed
The isometries of R3 that leave the origin fixed, forming the group O(3,R), can be categorized as follows:
- SO(3,R):
- identity
- rotation about an axis through the origin by an angle not equal to 180°
- rotation about an axis through the origin by an angle of 180°
- the same with inversion (x is mapped to −x), i.e. respectively:
- inversion
- rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis
- reflection in a plane through the origin
The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.
See also the similar overview including translations.
Read more about this topic: Point Groups In Three Dimensions
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