Re: 3DLDF

[Hans Aberg]

At 11:30 +0200 2004/08/14, Laurence Finston wrote:
>> The quaternions were created in order to describe 3D-rotations, but the
>> underlying math will be the same as if you work with matrices. So I am not
>> sure you want to get out there.
>
>I thought it might at least help me to spread the error more
>evenly.  I'd really appreciate it if you (or anyone else)
>could suggest something that might work better.

If you work directly with matrices, rotations are described by the
orthogonal matrices A, i.e., satisfying A*A^tr = 1. (This is essentially
what quaternions do in three dimensions.) One can describe them using
trigonometry.

If one entirely wants to avoid alignment problems when working with floats,
then  one should ideally work with only rational matrices. But it will in
general not be possible with such matrices, for example a 45 degree
rotation requires square roots. (If you want to entirely exclude some
rotations, so that only rational matrices are possible, then in two
dimensions this leads to working with Pythagorean numbers (a, b, c)
generated by the well known formula a = p^2 - q^2, b = 2pq, c = p^2 + q^2.)

But I am surprised that double's, if implemented using IEEE 64-bit floats,
causes alignment problems (about sixteen decimals accuracy). The accuracy
should be sufficient for use in graphics, I think. (There is also the GNU
GMP numerical multiprecision package one can experiment with, to see what
accuracy is needed.) But graphics experts most likely already know how to
solve such problems.

  Hans Aberg