Posted by B. Douglas Ward on March 28, 2001 at 09:33:48:
In Reply to: Re: Matrix file posted by B. Douglas Ward on March 27, 2001 at 17:09:55:
Phoebe:
Just for the record, let me emphasize that there is nothing "wrong" with a
non-orthogonal transformation. Some of my best friends use non-orthogonal
transformations. However, program 3drotate will only accept an orthogonal
matrix, which motivates the following:
Your original 3x3 matrix is almost orthogonal. So, perhaps it would be possible
to "perturb" the matrix elements to get an orthogonal matrix that is close to
the original matrix. To this end, I applied the Gram-Schmidt orthogonalization
process (other numerical methods may be better) to the columns of your original
3x3 matrix, and obtained:
0.998531 -.054177 -.000001
0.052336 0.964601 0.258469
-.014003 -.258089 0.966020
The above matrix is orthogonal, and the elements are somewhat close to the
elements of your original matrix.
Warning: There is no guarantee that the orthogonalized matrix will be "close"
to the original matrix.
Appending the original translation vector, we obtain:
0.998531 -.054177 -.000001 -3.0049
0.052336 0.964601 0.258469 20.682
-.014003 -.258089 0.966020 7.2701
I do know that 3drotate will work with the above input. Whether the output
is acceptable is up to you.
Doug Ward