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From: robert@victoria.esd.sgi.com (Robert Skinner)
Newsgroups: comp.graphics
Subject: Re: Errors in rotations
Keywords: ROTATIONS
Message-ID: <2496@odin.SGI.COM>
Date: 5 Jan 90 19:25:09 GMT
References: <47d784ba.20b6d@apollo.HP.COM> <1990Jan2.163405.24094@ibmpcug.co.uk>
Sender: news@odin.SGI.COM
Reply-To: robert@victoria.esd.sgi.com (Robert Skinner)
Organization: Silicon Graphics, Inc.
Lines: 57

In article <47d784ba.20b6d@apollo.HP.COM>, oj@apollo.HP.COM (Ellis
Oliver Jones) writes:
> I faced this problem some years ago on an E&S Picture System II.
> I used a compromise solution (hack) as follows.
> (1) The rotation matrix was kept separate, and was not composed
>     with translation or perspective matrices.
> (2) A good rotation matrix (as opposed to a malformed one) is
>     unitary.  This means that its determinant is one, obviously,
>     but it also means that each of the rows taken by itself
>     is a unit vector.  Ditto for the columns.
> (3) Every nth rotation cycle (1<n<20, empirically determined),
>     perform a normalization step.  Adjust the rotation matrix
>     such that each of the rows or columns, considered separately,
>     are unit vectors.  This works well if you alternate
>     between rows and columns in successive normalization
>     steps.  If you just normalize columns (or just rows)
>     the rotation will deteriorate into a skew.  In practice
>     the alternation prevents this deterioration.
> (4) The implementation of the normalization step was
>     carefully coded, using integer multiplies and
>     a decent integer Newton-Raphson-method square root.
>     See earlier postings in this newsgroup for some ideas.
> 
> /Ollie Jones (speaking for myself, not necessarily for HP Apollo, nor
for UCSF's
>              computer graphics lab, where I worked when I did this).

Ollie is correct on (1), (2), and (4), but step (3) depends on luck 
for the corrections applied to rows to correctly cancel out the
corrections applied to columns.
 
Remember that rotation matrices are not only unitary, but are
also orthogonal -- which means that each row (column) vector is
orthogonal to all other row (column) vectors.  You can use this fact
to get a correct rotation matrix without switching between rows
and columns.   Here's how:

starting with the first row vector (which I will call X, because
this is what a unit vector on the X axis gets transformed into), 
normalize it to obtain X'.  Then obtain the correct Z direction by 
finding the cross product of X' and Y and normalizing:
	Z' = norm( X' x Y )

Now you can find the correct Y by using the cross product again:
	Y' = Z' x X'
(no need to normalize, because X' and Z' are unit length and
orthogonal).  The matrix is now a real rotation matrix, unitary
and orthonormal.

You still need a fast square-root for the normalization.
Of course, this sort of assumes that X is correct each time.
You could start with Y or Z other times, but my guess is that
you probably won't notice any difference in sticking with X first,
because the matrix is now correct.

Robert Skinner
robert@sgi.com