Talk:Euler's theorem (rotation)

What is a rotation?

As I understand the first sentence, a rotation is defined to be "a motion of the rigid body that leaves at least one point of the body in place", but what is a rigid body motion? I think SE(3), i.e., all transformations of the form

${\mathbf {x}}\mapsto {\mathbf {Rx}}+{\mathbf {b}}$

with R in SO(3), however that does not seem to be what is meant in the article. -- Jitse Niesen 10:50, 14 May 2009 (UTC)

Yes, when b = 0 it is a rotation, provided R is an orthogonal matrix. When R = E it is a pure translation. I thought that "rigid body motion" would not have to be defined. See also Rotation matrix where I wrote the same (I'm still working on the latter). --Paul Wormer 11:23, 14 May 2009 (UTC)

But there are combinations of rotations and translations that leave points of the body in place. For instance, take

${\mathbf {x}}\mapsto {\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}{\mathbf {x}}+{\begin{bmatrix}1\\0\\0\end{bmatrix}}.$

This transformation leaves the point (1/2, 1/2, 0) in place, but it's not a rotation. So I think it's wrong to define a rotation as "a motion of the rigid body that leaves at least one point of the body in place". -- Jitse Niesen 16:47, 14 May 2009 (UTC)