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
Famous quotes containing the words leave, origin and/or fixed:
“Let us leave pretty women to men devoid of imagination.”
—Marcel Proust (1871–1922)
“We have got rid of the fetish of the divine right of kings, and that slavery is of divine origin and authority. But the divine right of property has taken its place. The tendency plainly is towards ... “a government of the rich, by the rich, and for the rich.””
—Rutherford Birchard Hayes (1822–1893)
“The aim of every artist is to arrest motion, which is life, by artificial means and hold it fixed so that a hundred years later, when a stranger looks at it, it moves again since it is life. Since man is mortal, the only immortality possible for him is to leave something behind him that is immortal since it will always move. This is the artist’s way of scribbling “Kilroy was here” on the wall of the final and irrevocable oblivion through which he must someday pass.”
—William Faulkner (1897–1962)