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I see that the discussion is shifting somewhat.
<p>Clifford J Mugnier wrote:
<blockquote TYPE=CITE>Gentlemen:
<p> Nanometer precision is meaningless if the transformation
does not
<br>produce the published and legally legislated results of a nation's
Grid
<br>system. For that reason, every country has an explicit formula
published
<br>along with a truncation to an infinite series. That, and only
that
<br>transformation is correct. Adding additional terms to allow
<br>transformations to be "precise" to greater distances from the projection
<br>origin is WRONG!</blockquote>
Major incorrect issue here is that changing computations does not
<br>necessarily change range of projection. In the case of improving
the
<br>meridinal distance computation of proj. the precision of the calculation
<br>was improved over the entire N-S range.
<blockquote TYPE=CITE> The only time "cleverness"
is allowed in transformations in
<br>association with National Legal Coordinate Systems is for the inverse
case
<br>where one goes from Grid to Geographic. In that case, the published
<br>formulae for the inverse case may be inadequate to allow one to obtain
the
<br>original result of the direct case. In those cases, and only
in those
<br>cases, one may use additional terms (7th, 8th, ... etc. derivatives),
or an
<br>iterative procedure to allow perfect "return" to the original coordinate
<br>when using the specific series truncation mandated by that nation for
the
<br>direct transform.</blockquote>
I should think that it should be the obligation of the soverign athority
to
<br>define the equations for <b><i>both</i></b> the forward and inverse
case.
<blockquote TYPE=CITE>There are specific truncations for Transverse Mercator
that include
<br>Gauss-Conform, Gauss-Schreiber, Gauss-Boaga, Gauss-Li, Gauss-Krüger,
etc.
<br>For Oblique Mercator there are Hotine, Laborde, and Rosenmund.
For Lambert
<br>there's several, and there's several for Oblique Stereographic, etc.,
etc.</blockquote>
In the case of the transverse mercator the different nomenclatures
often define
<br>different math and are not simply respecification of the limits of
series powers and
<br>coefficient evaluations. Same for the oblique. Simply tweeking
the basic Gauss-Kruger
<br>evaluation will not creat the numbers for some of the other variants.
<blockquote TYPE=CITE>This is not bean-counting, it's Applied Geodesy.
Applied Geodesy is what
<br>countries use in their legal coordinate systems for defining their
<br>international boundaries, their private property boundaries, their
<br>national-provincial boundaries, etc. It is used everywhere, it's
not just
<br>theory, and it's damned difficult to research. But it's there,
it exists,
<br>and it ain't theory.</blockquote>
I am somewhat bewildered by the above. All projections that I am
aware of are
<br>born out of mathematcal theory. What has happened in some cases
is that the
<br>theoretical definition varies in some cases and the practical computational
process
<br>has been compromised in others. Even sloppy derivation
(ie. sin(n x)) can be
<br>found in others.
<p>One case I was familiar with was the Swiss transverse mercator.
There is
<br>no way to tweek Gauss-Kruger to do the computations as the mathematical
<br>development and surface is quite different. Saying "transverse
mercator" does not define
<br>a unique surface tramsformation.
<blockquote TYPE=CITE>Single precision is rarely useful except on 64-bit
machines; a machine's
<br>episilon or internal precision is nice to know, but you have to match
the
<br>legal system for "it" to be correct. Arguing about the number
of digits
<br>the semi-major axis is published to, and using that as the justification
<br>for computational precision and significant digits is specious in this
<br>context. When your young son in uniform is on a frontier and
staring down
<br>the barrel of a cannon, the defense of your country's border is a matter
of
<br>legalities discussed by diplomats and defined by specific truncations
of
<br>series. World War I was a prime example of such ignorance by
the U.S. when
<br>the U.S.Coast & Geodetic Survey incorrectly added terms to the
formulae for
<br>the Lambert Conic in the Nord de Guerre Zone of France and Belgium.
The
<br>current standard is computational precision to a tenth of a millimeter
for
<br>the direct transform AS PUBLISHED BY A SOVERIGN NATION, and the inverse
<br>transform must "return" to the original geographic coordinates.</blockquote>
Presumably the above material published by the defining authority.
In my
<br>experience I have found it damned difficult, if not impossible, to
find such information.
<br>Even in the US I couldn't give you a reference for official US TM and
other projections
<br><b><i>and</i></b> a statement of the accuracy of conversion.
Just to find the precision of GK as
<br>given by Snyder and others, I went through a process of numerical integation
of
<br>various terms using extended precision (25 digits) computation.
I have never seen anyone
<br>else make a statement of computational accuracy other than "it was
good enough for
<br>government work."
<blockquote TYPE=CITE>My two cents.</blockquote>
In conclusioin, I find that map projection usage is in a pretty sorry state
when
<br>discussing precission of the product. Secondly, I do not consider
projections
<br>as part of "geodetic" subdiscipline but merely a mathematical
art of converting surfaces
<br>between planar and spherical/elliptical objects. I see geodesists
discussing
<br>precision a great deal when dealing with the earth's shape but nary
a whisper
<br>about cartographic projections.</html>