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