[OSRS-PROJ] Significant digits in parameters
Gerald Evenden
gerald.evenden at verizon.net
Sun Jul 28 16:21:30 EDT 2002
I see that the discussion is shifting somewhat.
Clifford J Mugnier wrote:
> Gentlemen:
>
> Nanometer precision is meaningless if the transformation does not
> produce the published and legally legislated results of a nation's Grid
> system. For that reason, every country has an explicit formula published
> along with a truncation to an infinite series. That, and only that
> transformation is correct. Adding additional terms to allow
> transformations to be "precise" to greater distances from the projection
> origin is WRONG!
Major incorrect issue here is that changing computations does not
necessarily change range of projection. In the case of improving the
meridinal distance computation of proj. the precision of the calculation
was improved over the entire N-S range.
> The only time "cleverness" is allowed in transformations in
> association with National Legal Coordinate Systems is for the inverse case
> where one goes from Grid to Geographic. In that case, the published
> formulae for the inverse case may be inadequate to allow one to obtain the
> original result of the direct case. In those cases, and only in those
> cases, one may use additional terms (7th, 8th, ... etc. derivatives), or an
> iterative procedure to allow perfect "return" to the original coordinate
> when using the specific series truncation mandated by that nation for the
> direct transform.
I should think that it should be the obligation of the soverign athority to
define the equations for both the forward and inverse case.
> There are specific truncations for Transverse Mercator that include
> Gauss-Conform, Gauss-Schreiber, Gauss-Boaga, Gauss-Li, Gauss-Krüger, etc.
> For Oblique Mercator there are Hotine, Laborde, and Rosenmund. For Lambert
> there's several, and there's several for Oblique Stereographic, etc., etc.
In the case of the transverse mercator the different nomenclatures often
define
different math and are not simply respecification of the limits of series
powers and
coefficient evaluations. Same for the oblique. Simply tweeking the basic
Gauss-Kruger
evaluation will not creat the numbers for some of the other variants.
> This is not bean-counting, it's Applied Geodesy. Applied Geodesy is what
> countries use in their legal coordinate systems for defining their
> international boundaries, their private property boundaries, their
> national-provincial boundaries, etc. It is used everywhere, it's not just
> theory, and it's damned difficult to research. But it's there, it exists,
> and it ain't theory.
I am somewhat bewildered by the above. All projections that I am aware of are
born out of mathematcal theory. What has happened in some cases is that the
theoretical definition varies in some cases and the practical computational
process
has been compromised in others. Even sloppy derivation (ie. sin(n x)) can be
found in others.
One case I was familiar with was the Swiss transverse mercator. There is
no way to tweek Gauss-Kruger to do the computations as the mathematical
development and surface is quite different. Saying "transverse mercator" does
not define
a unique surface tramsformation.
> Single precision is rarely useful except on 64-bit machines; a machine's
> episilon or internal precision is nice to know, but you have to match the
> legal system for "it" to be correct. Arguing about the number of digits
> the semi-major axis is published to, and using that as the justification
> for computational precision and significant digits is specious in this
> context. When your young son in uniform is on a frontier and staring down
> the barrel of a cannon, the defense of your country's border is a matter of
> legalities discussed by diplomats and defined by specific truncations of
> series. World War I was a prime example of such ignorance by the U.S. when
> the U.S.Coast & Geodetic Survey incorrectly added terms to the formulae for
> the Lambert Conic in the Nord de Guerre Zone of France and Belgium. The
> current standard is computational precision to a tenth of a millimeter for
> the direct transform AS PUBLISHED BY A SOVERIGN NATION, and the inverse
> transform must "return" to the original geographic coordinates.
Presumably the above material published by the defining authority. In my
experience I have found it damned difficult, if not impossible, to find such
information.
Even in the US I couldn't give you a reference for official US TM and other
projections
and a statement of the accuracy of conversion. Just to find the precision of
GK as
given by Snyder and others, I went through a process of numerical integation of
various terms using extended precision (25 digits) computation. I have never
seen anyone
else make a statement of computational accuracy other than "it was good enough
for
government work."
> My two cents.
In conclusioin, I find that map projection usage is in a pretty sorry state
when
discussing precission of the product. Secondly, I do not consider projections
as part of "geodetic" subdiscipline but merely a mathematical art of
converting surfaces
between planar and spherical/elliptical objects. I see geodesists discussing
precision a great deal when dealing with the earth's shape but nary a whisper
about cartographic projections.
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