[Proj] Computing the Lambert conformal conic projection accurately

[strebe]

Oscar:

Not to answer the question about proj specifically, since I do not know, but to talk about the Bonne in general: The Bonne is highly unstable when the standard parallel is near the equator. The formulas must get recast from the usual forms, and, once recast, must further get developed as a (very complicated) series for latitudes close to the equator. If proj is not doing this then gross errors should be expected.

Why do you say that the difference probably cannot be attributed to instabilities?

The Bonne with a standard parallel of 0° is identical to the sinusoidal.

Regards,
— daan Strebe


On Nov 21, 2010, at 3:24:01 AM, OvV_HN <ovv at hetnet.nl> wrote:

There is probably a more serious problem in one of the other projections, 
namely the Bonne projection.
If lat_0 is close to 0, the Sinuoidal projection is taken.
If one compares in a limiting case the northing, produced by the Sinusoidal, 
and for the same set of parameters the northing, produced by the Bonne 
projection, a difference of perhaps 300 meters will be obtained. This 
observation is valid for both the ellipsoidal and the spherical projections.
This difference can probably not be attributed to instabilities in the 
calculation or a difference between a limiting lat_0 (of 1e-10 deg) and 
zero.
Can somebody enlighten me? Does the Sinusoidal projection differ a bit too 
much from the Bonne (but no one cares)?
Is there something wrong in the proj/libproj code?
Or did I do something wrong?

My test case:

Bessel 1841 ellipsoid
lat=40d 32m; lon=-7d 16m;
lat0=1e-10; lon0=-8.131906111111112;
x0=20000; y0=10000; [meters]
bonne ellipsoidal projection
Result: x = 93299.284; y = 4497999.829; [meters]

for lat0=0 the Sinusoidal projection is taken.
Result: x = 93299.284; y = 4498299.575;
Difference in y: 299.75 m

Oscar van Vlijmen


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