[Proj] Discovery: libproj4 stmerc = French Gauss-Laborde projection
Gerald I. Evenden
gerald.evenden at verizon.net
Tue Jun 13 11:22:54 EDT 2006
On Tuesday 13 June 2006 4:51 am, Oscar van Vlijmen wrote:
> Going complex is not needed. Those few sums of complex sines can easily be
> computed doing Re and Im separately in reals.
> NTG 73 describes 3 types of Gauss-Laborde projections. One of them, the
> "sphère de courbure", is equal to stmerc.
After looking more closely last night I began to come to that conclusion. I
am somewhat curious about the term "sphère de courbure" which an online
translator gives: "curve of sphere." Also, I am not yet quite sure what
simplifies in the équatoriale and bitangente cases.
> If you throw a lot of lat, lon, lat0, k0 &c parameters to both and you get
> the same results everywhere, they must be the same. This is no rigorous
> proof of course.
> NTG 76 describes algorithms for the transverse Mercator projection, using
> complex math. One of the advantages of this route is that you get very good
> results for large differences in lon-lon0.
> For instance, if you go 30 degrees away from the central meridian (lon0),
> at latitude 10 degrees, you'll find that tmerc departs around 60 m in x and
> around 30 m in y from the exact value.
> Routines like the French are only microns away from the exact values.
> Millimeter accuracy is still available to at least lon-lon0 = 70 degrees,
> and beyond that, only for smaller values of the latitude (say <30 deg) the
> function gets worse.
There is a solution out there that goes all the way but uses functions that
are hard to find and/or need to be developed. One fellow supposedly sped up
the solution but I could not reproduce his results.
> At least three geodetic services use routines approximating the exact TM
> better than tmerc, DMA/NIMA/NGA and the like:
> 1) French IGN (see above)
> 2) Swedish Lantmäteriet
> 3) Finnish JHS
I timed out trying to get to the above url. I will try again later.
> Each follow a slightly different route, but the differences in the results
> are small.
> For small values of lon-lon0, say less than 6 degrees, tmerc et alii
> perform very well, so nothing can be gained using the above mentioned TMs.
One might question the practical need to go beyond 6 degrees.
BTW: what did you use to get the "exact" values?
> Let's give a testpoint:
> WGS84 ellipsoid
> lat=40; lon=70; lon0=0; lat0=0; x0=5e5; y0=0; k0=0.9996;
> x,y exact: 6289992.60347, 7531297.26735 m
> The French routines do: 6289992.60342, 7531297.26746
> The Finnish routines give: 6289992.60323, 7531297.26746
> By the way, the Finnish routines compute a meridian convergence of 58.289
> deg and a point scale factor of 1.43738.
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
Ambrose Bierce, The Devil's Dictionary
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