Path: utzoo!attcan!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!samsung!sdd.hp.com!decwrl!bacchus.pa.dec.com!shlump.nac.dec.com!ryn.esg.dec.com!allvax!jroth
From: jroth@allvax.dec.com (Jim Roth)
Newsgroups: comp.graphics
Subject: Re: 4-Space Basis Rotation Matrices
Summary: Correction with respect to Euler angles
Message-ID: <2498@ryn.esg.dec.com>
Date: 12 Aug 90 17:57:33 GMT
Sender: guest@ryn.esg.dec.com
Organization: Digital Equipment Corporation
Lines: 54


In article <2494@ryn.esg.dec.com>, jroth@allvax.dec.com (Jim Roth) writes...
> 
>In article <1279@enuxha.eas.asu.edu>, hollasch@enuxha.eas.asu.edu (Steve Hollasch) writes...
>> 
>>    I'm writing a 4D wireframe-viewer and have run into the problem of
>>generating rotation matrices for the 4D viewpoint.  What are the basis
>>rotation matrices in 4D?

[ stuff deleted ... ]

I just wanted to make a correction in my reply about Euler angles.  Though it is
true that there are 6 pairs of coordinate axes in 4 space, the correct
composition of generalized Euler angle matrices does not involve matrices
actually containing rotations in all these planes.

Consider an n-1 dimensional rotation which fixes the n'th coordinate axis.
This is a general rotation in n-1 space.  If this is followed with a
rotation which takes the n'th coordinate axis to an arbitrary point on the
n-sphere you can inductively generate arbitrary Euler angle rotations.

A systematic way to do this is to apply successive plane rotations in
the (k,k-1) plane, then the (k-1,k-2) plane, ... then the (1,2) plane
for k = 2 to n.  Of course, this adds up to n(n-1)/2 matrices total.

Thus in 4-D the series would be (assuming you multiply with column vectors
on the right):

    | C -S  0  0 | | 1  0  0  0 | | 1  0  0  0 |
    | S  C  0  0 | | 0  C -S  0 | | 0  1  0  0 |
    | 0  0  1  0 | | 0  S  C  0 | | 0  0  C -S |
    | 0  0  0  1 | | 0  0  0  1 | | 0  0  S  C |

      (1,2) plane    (2,3) plane    (3,4) plane

    | C -S  0  0 | | 1  0  0  0 |
    | S  C  0  0 | | 0  C -S  0 |
    | 0  0  1  0 | | 0  S  C  0 |
    | 0  0  0  1 | | 0  0  0  1 |

      (1,2) plane    (2,3) plane

    | C -S  0  0 |
    | S  C  0  0 |
    | 0  0  1  0 |
    | 0  0  0  1 |

      (1,2) plane

   That is, your rotation matrix would be

	(1,2) * (2,3) * (3,4) * (1,2) * (2,3) * (1,2)

    - Jim