VCV OF TRANSFORMED COORDINATES |
When known parameters are used to transform coordinates the variance-covariance
(VCV) matrix of these coordinates should also be transformed. The VCV of
transformed coordinates can be determined by applying the Law of Propagation
of Variances. The following assumes small rotation angles, but similar equations
could be derived for the full R matrix if required. Assuming
the following transformation model (eqn (11.1-1)):
 |
(11.1-6) |
 |
(11.1-7) |
In VCVP (the 7x7 VCV
matrix of the parameters) the transformation parameters are in the order:
s, 
, 
, 
, Tx, Ty, Tz. VCVXYZ'A' and VCVXYZ'B'
are the 3x3 VCV matrices of the coordinates of the point being transformed.
Similarly, the VCV matrix of a network of transformed coordinates XB1,YB1,ZB1 to XBn,YBn,ZBn can be determined as follows:
 |
(11.1-8) |
 |
VCVXYZ'An' and VCVXYZ'Bn' are 3nx3n matrices.
Thus the VCV matrix of the transformed coordinates VCVXYZ'Bn' is a combination of the VCV matrices of the original coordinates VCVXYZ'An' and the transformation parameters used VCVP.
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© Chris Rizos, SNAP-UNSW, 1999