From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Rotations about 2 or 3 axis
Date: 29 Jul 1996 16:28:31 GMT
In article <838334834.3918.0@jacy.demon.co.uk>, john@jacy.demon.co.uk (John Cranmer) writes:
> Using the normal x, y and z axis system it is possible to orientate a
> plane in any position by rotating about these three axes.
> What I would like to know is;
> Is it always posssible to reduce the three rotations to a pair of
> rotations about two of the axes, and if so is there a general formula
> to calculate the necessary angles.
In article <4tbbep$o02@nntp.ucs.ubc.ca>,
Robert Israel wrote:
>No. The rotation group SO(3) is 3-dimensional.
This is correct, but does not imply that it can't be generated by just
two one-dimensional families, which is how I understood the
question. The rotation by an angle t around the first axis, given in
matrix form as
[ 1 0 0 ]
[ 0 cos(t) -sin(t) ]
[ 0 sin(t) cos(t) ]
may be represented as a product of rotations around the other two axes.
It's R S R^(-1) where R =
[ 0 0 -1 ]
[ 0 1 0 ]
[ 1 0 0 ]
(this is a 90-degree rotation around the second axis) and S =
[ cos(t) sin(t) 0 ]
[ -sin(t) cos(t) 0 ]
[ 0 0 1 ]
(this is a rotation by an angle -t around the third axis). So if you
believe that the three families of rotations generate all of SO(3),
then it follows the last two families alone also generate it.
dave