Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!wuarchive!mailrus!ncar!asuvax!enuxha!hollasch
From: hollasch@enuxha.eas.asu.edu (Steve Hollasch)
Newsgroups: comp.graphics
Subject: 4-Space Basis Rotation Matrices
Keywords: 4D 4-Space four dimension rotation
Message-ID: <1279@enuxha.eas.asu.edu>
Date: 9 Aug 90 20:44:08 GMT
Organization: Arizona State University
Lines: 41


    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?  The 4-space viewpoint will be on the surface
of a hypersphere, and I want a set of basis rotations that can rotate
the point to any other point on the surface of that hypersphere.  In
addition, the rotation needs to be parametric, since I'll be tying the
rotation angle to the cursor for a smooth rotation.  Hence, I don't
want reflections or other improper rotations.

    I think that two rotation matrices are sufficient in 3-space;
consider spherical coordinates with a fixed radial length.  If this is
true, I'd also like to know what the analogous 4-space rotation
matrices would be.

    So how do you find the set of 4-space rotation matrices?  I can't
think of a way to rotate about the basis vectors in 4-space, since
traditional rotation matrices deal with rotations in a plane.  One
suggested approach is to include each 3-space rotation matrix in a 4x4
matric for each row/column pair replaced with the identity row/column,
e.g.  ( S = sin(theta), C = cos(theta) ):

    +-          -+   +-          -+   +-          -+   +-          -+
    | 1  0  0  0 |   | 1  0  0  0 |   | 1  0  0  0 |   | C  0 -S  0 |
    | 0  C  S  0 |   | 0  C  0 -S |   | 0  1  0  0 |   | 0  1  0  0 |
    | 0 -S  C  0 |   | 0  0  1  0 |   | 0  0  C  S |   | S  0  C  0 |
    | 0  0  0  1 |   | 0  S  0  C |   | 0  0 -S  C |   | 0  0  0  1 |
    +-          -+,  +-          -+,  +-          -+,  +-          -+, 

    and so on, for a total of twelve matrices.

    Are these twelve matrices sufficient to parametrically rotate a 4D
viewpoint on the 4-sphere to any other viewpoint on the 4-sphere?  Are
they all _necessary_ (like only needing two rotations in 3-space)?

    Any comments or literature references?
-- 
______________________________________________________________________________
Steve Hollasch                       Arizona State University / Tempe, Arizona
hollasch@enuxha.eas.asu.edu  /  uunet!mimsy!oddjob!noao!asuvax!enuxha!hollasch
                          Is this my opinion yet?