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:
“I conjure thee, and all the oaths which I
And thou have sworn to seal joint constancy,
Here I unswear, and overswear them thus,
Thou shalt not love by ways so dangerous.
Temper, O fair Love, love’s impetuous rage,
Be my true Mistress still, not my feign’d Page;
I’ll go, and, by thy kind leave, leave behind
Thee, only worthy to nurse in my mind
Thirst to come back;”
—John Donne (1572–1631)
“In the woods in a winter afternoon one will see as readily the origin of the stained glass window, with which Gothic cathedrals are adorned, in the colors of the western sky seen through the bare and crossing branches of the forest.”
—Ralph Waldo Emerson (1803–1882)
“It is not merely the likeness which is precious ... but the association and the sense of nearness involved in the thing ... the fact of the very shadow of the person lying there fixed forever! It is the very sanctification of portraits I think—and it is not at all monstrous in me to say ... that I would rather have such a memorial of one I dearly loved, than the noblest Artist’s work ever produced.”
—Elizabeth Barrett Browning (1806–1861)