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From: oj@apollo.HP.COM (Ellis Oliver Jones)
Newsgroups: comp.graphics
Subject: Re: Errors in rotations
Keywords: ROTATIONS
Message-ID: <47d784ba.20b6d@apollo.HP.COM>
Date: 4 Jan 90 23:49:00 GMT
References: <1990Jan2.163405.24094@ibmpcug.co.uk>
Sender: root@apollo.HP.COM
Reply-To: oj@apollo.hp.com
Organization: HP Apollo Computer, Chelmsford, MA
Lines: 29

In article <1990Jan2.163405.24094@ibmpcug.co.uk> peterl@ibmpcug.co.uk (Peter Leaback) writes:
>...I define my rotations as a 3x3 matrix containing 16 bit
>fixed point numbers from 1 to -1.  ... after a few
>thousand rotations,the errors that occur in combining rotations has
>malformed the matrix....

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).