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:
“Come leave your tears: a brief farewell. The beast
With many heads butts me away.”
—William Shakespeare (1564–1616)
“Each structure and institution here was so primitive that you could at once refer it to its source; but our buildings commonly suggest neither their origin nor their purpose.”
—Henry David Thoreau (1817–1862)
“I, who travel most often for my pleasure, do not direct myself so badly. If it looks ugly on the right, I take the left; if I find myself unfit to ride my horse, I stop.... Have I left something unseen behind me? I go back; it is still on my road. I trace no fixed line, either straight or crooked.”
—Michel de Montaigne (1533–1592)