Rotation in 3 dimensions about a point

[tut_einstein]

Hi,

I was wondering whether it's possible to define a rotation in 3 dimensions about the origin. Is it necessary to define an axis of rotation or would it be legal to say that you rotate abput the origin (like a phasor in 3 dimensions.)

Thanks!

 

[Tinyboss]

Every rotation in 3 dimensions leaves a line through the origin unchanged, and we call it the "axis of rotation". It's true in all the higher odd dimensions as well, although in that case there may not be a single uniquely defined axis for a given rotation.

In even dimensions, most rotations do not fix any points (except for the origin, of course).

 

[chiro]

Hey tut_einstein.

Since your origin of the reference frame corresponds to the real origin, no translation is needed. If this wasn't the case you would need to perform two translations before and after your rotation in terms of a linear composition (i.e. matrices to be pre and post multiplied)

For specifics of the rotation, take a look at this:

http://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle

 

[mathwonk]

a rotation is a specific distance preserving and orientation preserving linear transformation. Thus all its eigenvalues have length one and their product is 1.

In three - space the characteristic polynomial whose roots are the eigenvalues is a cubic with real coefficients. Thus it has at least one real root, either 1 or -1. Either way, the line spanned by the corresponding eigenvector is mapped into itself.

The case of a rotation is the one in which the real eigenvalue is 1, and the other two roots are complex conjugates. The only case when they are also real for a rotation, is an 180 degree rotation where they are both equal to -1. Thus for a rotation, the line spanned by the eigenvector with eigenvalue 1, is not only preserved but fixed pointwise.

If you define a "rotation" simply to mean an orientation preserving, distance preserving, linear map, then it is not hard to write down a matrix in even dimensions, which has orthogonal rows and determinant one, but no characteristic root equal to 1. Try it.

In fact you should be able to make the characteristic polynomial a power of X^2+1.