Just in case some are curious, there is the steps:
Lets use the following symbols in the remainding of this email:
[ ] denote an affine transform
< > denote the non-linear part of a projection
X' means the inverse of X
X*Y means "transform X followed by transform Y".
If X and Y are affine transforms, they can be
concatenated using matrix multiplication. I will
call the resulting matrix [XY] (this is NOT the
matrix multplication order - it is just notation
for this email).
Lets suppose that we have two projections that depend only on excentricity
(after we moved linear terms to the affines), for example Mercator or Transverse
Mercator.
[A] * <B> * [C]
and
[D] * <E> * [F]
Then going from the first to the second projection is:
([A] * <B> * [C])' * ([D] * <E> * [F])
= [C]' * <B>' * [A]' * [D] * <E> * [F]
= [C]' * <B>' * [A'D] * <E> * [F]
If [A'D] is not the identity transform (which is the case when changing between
two UTM zones because of the non-zero translation term in X), then we can't go
any further and we are done.
If [A'D] is the identity transform (which is the case when changing between two
Mercator projection having different scale factor or standard parallels - note
that the different scale factors appear in [C] and [F] affine transforms, not in
[A] or [D] transforms, then the expression can be simplified further:
= [C]' * <B>' * <E> * [F]
referencing code should be able to see that <B> and <E> are identical if there
is nothing else than the excentricity as non-linear parameters. So <B>' * <E> =
identity transform.
= [C]' * [F]
= [C'F]
It does not happen so often in practice, but we still have a few cases. And most
important, the non-linear part of projection formulas are simplier and handling
of ESRI's X_Scale, Y_Scale, XY_PlaneRotation parameters (used in Krovak
projection) and handling of South Orientated projection (e.g. Transverse
Mercator) are no longer a hack, since they fit well in affine transforms.
Martin
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