[Proj] RE: "Double Ellipsoid" error, reproduction

[Mikael Rittri]

Summary:
-------
I think I now agree with Noel on all hard facts. We just have 
to find an objective way to translate the hard facts to the 
appropriate emotion, on a scale of 

1: Shrug,
2: Regret,
3: Contempt,
4: Loathing,
5: Smite the infidels with protractors.

Seriously, I will just grumble a bit about Simpson's rule.

Then I will - in reply to Richard Greenwood - list some software
that do implement spherical projection formulas on geodata from 
ellipsoidal datums. 

Long text:
---------
Noel Zinn wrote: 
> I demonstrated how a user can extract geodesic distances from grid 
> coordinates and point scale factors using Simpson's Approximation.
> That's the way most surveyors work (I believe), not using (recovered) 
> geographicals and geodesic computations, though we may agree that's
> the best route.  

Yes, you gave an impressive example where Simpson's rule
gave submillimeter accuracy over a distance of 111 km. 
[ http://lists.maptools.org/pipermail/proj/2008-December/004104.html ]

> What user knows to take the WGS84 ratio of rho and nu (the radii of
> curvature in the meridian and prime vertical) with the Equatorial
> radius and apply those to the spherical Mercator point scales in
> order to find the real distortions in GMP?  

Okay, not many. On the other hand, how many users know how to use 
Simpson's rule in this situation?  From my own experience, our 
customers - jolly good chaps though they are, and perhaps reading 
this list - usually forget that there is a local scale factor 
to consider at all. So, for accuracy, we provide geodetic 
computations via geographic lon/lat. 

And Simpson's rule is just an approximation, good only for short distances.
If someone would use Simpson's rule to compute the geodetic distance 
from New York (let's say 74°W, 41°N exactly) to Istanbul (let's say 
29°E, 41°N exactly), on the WGS84 / Ellipsoid Mercator, I think they 
would get a distance of 8665924.07 m, which is very accurate - but 
it is the distance along the 41°N parallel circle. Because Simpson's 
rule cannot handle that geodetic lines will appear curved in Mercator.  
The true geodetic distance is only 8071386.81 m, so Simpson's rule 
have overestimated the distance by 7.4 percent. Which should be 
compared with the 0.67 percent that we have agreed is the maximal 
difference between Sphere Mercator and Ellipsoid Mercator.


Richard Greenwood wrote: 
> I've not seen software that allows different ellipsoids
> for the projection and datum. Does such software exist?

Yes. In Carmenta Engine, the design is like this: 
A coordinate reference system is represented by a RefSystem object, 
which consists of a GeodeticDatum and a Projection. Normally, 
the reference ellipsoid is specified as part of the GeodeticDatum,
and the Projection will use that ellipsoid.  However, the Mercator 
class has an extra parameter called Variant, whose value comes from
an enumeration type. 
*) A Mercator with Variant = #ellipsoid will (of course) use 
ellipsoid formulas on the ellipsoid defined by the GeodeticDatum.  
*) But a Mercator with Variant = #sphereEquatorialRadius will use 
spherical formulas on a sphere, whose radius equals the equatorial 
radius of the ellipsoid.  
*) A third possibility, Variant = #sphereNautMileIsMinute, will 
completely ignore the ellipsoid dimensions, and instead use a 
sphere whose radius is 6366707.0195 m, making 1 nautical mile 
equal to 1 arc minute. (This is an old Carmenta tradition.) 

This design is similar to the 2nd design mentioned by Melita Kennedy 
of ESRI [ http://lists.osgeo.org/pipermail/metacrs/2008-August/000144.html ]. 

GeoTools uses a somewhat different design 
[ http://lists.osgeo.org/pipermail/metacrs/2008-August/000143.html ].


Best regards
--
Mikael Rittri
Carmenta AB
Box 11354
SE-404 28 Göteborg
Visitors: Sankt Eriksgatan 5
SWEDEN
mikael.rittri at carmenta.com
www.carmenta.com