[OSRS-PROJ] Re: On the Ellipsoidal Transverse Mercator
Strebe at aol.com
Strebe at aol.com
Sat Oct 18 01:11:06 EDT 2003
Posted to the group for those interested:
Karsten Engsager privately supplied to me a table of coordinates generated by
his implementation of the Poder-Engsager algorithm for the ellipsoidal
transverse Mercator. His algorithm achieves high accuracy through a range far beyond
the usual series expansions for the UTM. (See his mail to the group posted 15
October 2003.) I am very pleased to find that my implementation of Wallis's
algorithm and Karsten implementation of Poder-Engsager agree to within a
millimeter accuracy across the entire range of values for which Engsager claims his
implementation is accurate. Every point that deviates between our computations
is marked in Engsager's table as an inaccuracy in his algorithm. Hence the
visual discrepancy that I reported earlier can be due only to display anomalies
rather then computational inaccuracies.
This is an excellent affirmation of Engsager's algorithm, of the algorithm he
used to check his algorithm, of Wallis's algorithm, and of my implementation
of Wallis's algorithm.
The only fly in all this ointment is that Engsager marks several values from
his computations as being inaccurate, yet my calculations agree with his to
millimeter accuracy. Possible explanations:
1) His standard for accuracy exceeds millimeter accuracy, despite that his
table is prepared only to millimeter accuracy;
2) His algorithm for measuring error is not entirely accurate;
3) Both algorithms fail in exactly the same way, despite working completely
differently.
(3) is vanishingly unlikely. I cannot comment on (1) or (2).
For Mr. Engsager's benefit, the spherical coordinates of the projected
coordinates which we compute to be identical but which he marks as inaccurate are:
(latitude, longitude)
50.0 90.0
70.0 40.0
60.0 30.0
50.0 30.0
50.0 20.0
50.0 10.0
as well as the negative counterparts. Engsager's table shows some deviant
values as "inaccurate" and others as "serious". I don't know what the dividing
line between them is. Most of the values in his table marked "inaccurate"
deviated from my calculations by mere millimeters. One value in particular, though,
deviated by 20cm on one axis and 8cm on the other, yet it was still only
marked "inaccurate" rather than "serious". That coordinate is
20.0 80.0
as well as the negative counterparts. I'm not sure of the significance of
this.
As a consequence of this short study I can definitely recommend
Poder-Engsager's algorithm within its recommended range on earth-like ellipsoids.
Regards,
daan Strebe
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