[OSRS-PROJ] Re: On the Ellipsoidal Transverse Mercator

[Strebe at aol.com]

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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