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 <israel@math.ubc.ca> 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