Re: Unitary matrix. - Rotation matrix

From: JEMebius (jemebius_at_xs4all.nl)
Date: 10/26/04 

Date: Tue, 26 Oct 2004 20:34:40 +0100

Dear mr Bagula, my thanks for your interest in my SO(4) work.

You propose to perform rotations in each of the 6 coordinate planes;
this is a most natural approach to obtain more general rotations than
just a rotation in a single plane. Actually this was Euler's approach in
studying N-dimensional rotations; however he failed to give a complete
proof that one can obtain == all == N-dimensional rotations about a
fixed point in this way.
In the 4D case Arthur Cayley and Van Elfrinkhof observed that one
obtains 4D rotations by means of quaternions, but both of them failed to
prove, like Euler, that == all == 4D rotations can be obtained by
quaternions. This gap is filled by Bouman's 1932 article and in an
entirely different manner by my paper.

In rotations in 3D space (SO(3)) the three Euler angles are well-known.
Several different conventions are in common use as regards the order of
rotations in the successive coordinate planes. All of them have 3
independent parameters. A rather deep theorem says that any
representation of SO(3) by means of 3 independent parameters has
singularities somewhere. It is the same for representations of SO(4) by
means of 6 independent parameters.

I checked that with the three angles of yaw, pitch and roll of aircraft
one can reach each attitude in 3D space in precisely one way, except the
vertical nose-up and nose-down attitudes, which can reached in
infinitely many ways.
This is because in vertical flight the roll and pitch axes coincide and
one cannot any longer distinguish roll and pitch angles from each other.
I never checked how this works for the 6 rotation angles in your letter.

In conclusion: indeed quaternions in SO(4) seem not necessary, but turn
out to be the only practical way to handle 4D rotations without
stumbling upon singularities sooner or later.

===== reply to =====

Roger Bagula wrote:

> That sure is the hard way: I muse six angles{a,b,c,d,e,f}
> and six unitary matrices: Det[mi]=1
> m1={{Cos(a),Sin(a),0,0},{-Sin(a),Cos(a),0,0},{0,0,1,0},{0.0.0.1}}
> m2={{Cos(b),0,-Sin(b),0},{0,1,0,0},{Sin(b),0,Cos(b),0},{0.0.0.1}}
> m3={{1,0,0,1},{0,Cos(c),Sin(c),0},{0,-Sin(c),Cos(0),0,},{0.0.0.1}}
> m4={{Cos(d),0,0,-Sin(d)},{0,1,0,0},{0.0.1.0},{Sin(d),0,0,Cos(d)}}
> m5={{1,0,0,0},{0.1.0.0}{0,0,Cos(e),-Sin(e)},{0,0,-Sin(e),Cos(e)}}
> m6={{1,0,0,0},{0,Cos(f),0,Sin(f)},{0,-Sin(f),0,Cos(f)},{0.0.0.1}}
> Those give rotations in the six SO(4) orthogonal directions and are
> each unitary.
> A product will, thus, be unitary.
> I hope I didn't make any typing mistakes. (That is in Mathematica
> array notation. )
> Using quaternions in SO(4) seems entirely unnecessary.
> It's not a proof , but it is the logical way to use SO(4) to get a
> unitary rotation matrix.

> JEMebius wrote:
>
>> A reply to "Unitary Matrix." by Kurda Yon
>>
>> Please look at http://www.xs4all.nl/~jemebius/So4.htm - In this paper
>> the general 4x4 rotation matrix is expressed in terms of 8 real
>> parameters with 2 relations. The formulas for the matrix elements in
>> my paper were already known to Leonhard Euler
>> (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html),
>> to Arthur Cayley
>> (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html),
>> to Lambertus van Elfrinkhof and presumably to many other mathematicians.
>> The new thing in my paper is my proof that this representation is
>> essentially unique. I guess that it is the second proof of this SO(4)
>> representation theorem in the literature. My proof directly leads to
>> computer implementation of 4D rotations.
>>
>> On general NxN rotation matrices:
>>
>> (1) Euler proposed a general formula for linear transformations of N
>> variables leaving the sum of their quadrates unchanged. (Problema
>> algebraicum ob affectiones prorsus singulares memorabile, 1770) This
>> formula involves
>> N.(N - 1)/2 angular parameters. Euler did not use the terminology of
>> N-dimensional orthogonal transformations because at that time
>> N-dimensional geometry was still unknown.
>>
>> (2) Consult the articles by Arthur Cayley on "skew determinants" and
>> "déterminants gauches", or the older graduate-level or
>> advanced-undergraduate-level literature on matrices, determinants,
>> 19th-century higher algebra and real and complex linear algebra.
>>
>> HTH: Johan E. Mebius
>
>
>

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