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From: jcs@crash.cts.com (John Schultz)
Newsgroups: comp.graphics
Subject: Re: Cumulative error in rotation matrices
Keywords: matrices, errors, aipdc@castle started this thread
Message-ID: <6002@crash.cts.com>
Date: 30 Nov 90 21:37:46 GMT
References: <7386@castle.ed.ac.uk>
Distribution: comp
Organization: Crash TimeSharing, El Cajon, CA
Lines: 37

In <7386@castle.ed.ac.uk> aipdc@castle.ed.ac.uk (Paul D. Crowley) writes:
[stuff deleted]
>My problem is that M is gradually getting further and further from being
>a rotation.  Since the value for cos(x) is not exact, the matrix A is
>not exactly a rotation but a rotation followed by some other (small)
>transformation. But since M is multiplied by one of these very often,
>these small transformations build up, and the image becomes skewed in
>lots of nasty ways.

>What I'd like to do is apply some sort of process to M every so often
>that brings it back into line, ie that finds the rotation which M is
>closest to and changes it into that.  But I have no idea what form what
>rotations will take.

>Incidentally, if it makes a difference, I lied a little bit up there: M
>is actually 4x4 since I'm operating on four-dimensional scenes. 
>Solutions for three dimensions would also be appreciated, though.

  If you consider your matrix as a set of orthonormal vectors, then all
you need to do is restore the angles and lengths of the vectors so
they are unit length and orthogonal to each other. There are many ways
to do this. The simplest (and quickest) way is to normalize the first
column of your matrix, then cross that vector with the next
column, normalize the result, then cross the first column with the one
you just normalized. The three resulting vectors make up your new matrix.
For a 4x4 matrix, extend the process. Also, using this method, you don't
need to do a complete matrix multiply, as the last column is completely
interpolated. The column you pick as first to normalize will have the least
angular error. I normalize cfi first, as the view is looking down z.
  Other methods include just renormalizing rows and columns on alternate
passes, or a more complex (but more accurate method) is to find the matrix
that is closest to the current matrix and is also orthonormal. The methods
in this paragraph are described completely in "Graphics Gems", edited by
Andrew Glassner [Academic Press].


  John