Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!sdd.hp.com!usc!apple!sun-barr!newstop!sun!peregrine!falk
From: falk@peregrine.Sun.COM (Ed Falk)
Newsgroups: comp.graphics
Subject: Re: 4-Space Basis Rotation Matrices
Message-ID: <140587@sun.Eng.Sun.COM>
Date: 13 Aug 90 23:00:52 GMT
References: <2498@ryn.esg.dec.com> <1538@seti.inria.fr>
Sender: news@sun.Eng.Sun.COM
Organization: Sun Microsystems, Mt. View, Ca.
Lines: 47

In article <1538@seti.inria.fr> hussein@bora.inria.fr (Hussein Yahia) writes:
>
>
><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?
>
>  If n = 4, the theorem implies that EVERY rotation matrix in 4-space can
>be put in the following form:
>
>
>                 cos(a)       -sin(a)       0         0
>                 sin(a)        cos(a)       0         0
>                   0             0         cos(b)    -sin(b)
>                   0             0         sin(b)     cos(b)
>

I think a simpler way to think of it is this:  the idea that rotations
are around an *axis* is incorrect.  It just happens to work in
three dimensions.  In fact, rotations occur within a *plane*.  In
4D, there are six planes (XY, XZ, XW, YZ, YW, ZW) and the rotations
through those various planes are

     cos(a)  -sin(a)	0	0
     sin(a)   cos(a)	0	0
	0	0	1	0	XY plane
	0	0	0	1

     cos(a)	0   -sin(a)	0
	0	1	0	0	XZ plane
     sin(a)	0    cos(a)	0
	0	0	0	1

     cos(a)	0	0    -sin(a)
	0	1	0	0	XW plane
	0	0	1	0
     sin(a)	0	0     cos(a)

and so on.

	-ed falk, sun microsystems -- sun!falk, falk@sun.com
	"What are politicians going to tell people when the
	Constitution is gone and we still have a drug problem?"
			-- William Simpson, A.C.L.U.