Discovery: libproj4 stmerc = French Gauss-Laborde projection

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Discovery: libproj4 stmerc = French Gauss-Laborde projection

OvV_HN
stmerc is a projection in the libproj4 package. There is no documentation.
Mr. Evenden mentioned it on this discussion list on 2004-03-26.
Recently I found a connection with a projection in real use, the
Gauss-Laborde type "sphère de courbure" (probably meaning: the projection
sphere has a radius equal to the radius of curvature at the point of
origin).
What is the connection? They are the same!

There is documentation available at the French Institut Géographique
National, publication "Notes Techniques", NT/G 73.
<http://www.ign.fr/affiche_rubrique.asp?rbr_id=1700&lng_id=FR#68096>
<http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>

This publication gives the algorithms of two more Gauss-Laborde type
projections, one with a "sphère equatoriale" and one with a "sphère
bitangente".
The "sphère de courbure" projection was or is still in use for Réunion.
I have no idea where the other two Labordes were used for.

More background information in:
ENSG-IGN, Didier Bouteloup, Cours de Géodésie, Chapitre 3: §4.5.d
"Représentation de Gauss-Laborde"
<http://www.ensg.ign.fr/~bouteloup/www/wwwfad/site_fad/pdf/index_pdf.htm>
<http://www.ensg.ign.fr/~bouteloup/www/wwwfad/site_fad/pdf/chap3.pdf>

Example:
International ellipsoid;
lat=-21; lon=55.5;
lat0=-21d7m; lon0=55d32m; x0=1.6e5; y0=5e4; k0=1;
x,y Laborde type 1 = stmerc = 156534.17713, 62916.92507



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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
On Monday 12 June 2006 5:47 am, Oscar van Vlijmen wrote:
> stmerc is a projection in the libproj4 package. There is no documentation.
> Mr. Evenden mentioned it on this discussion list on 2004-03-26.
> Recently I found a connection with a projection in real use, the
> Gauss-Laborde type "sphère de courbure" (probably meaning: the projection
> sphere has a radius equal to the radius of curvature at the point of
> origin).
> What is the connection? They are the same!

The 's' in smerc comes from Schreiber Mercator fir which I have found very
little documentation.  Snyder refers to it as an alternative transverse
Mercator but gives no further useful information.

The libproj4 smerc is a Gauss transformation to the sphere, radius at lat_0,
and spherical transverse Mercator projection.

I cannot add anything about French usage but it has nothing to do with the
Laborde projection used in Madagascar.

If anyone has mathematic documentation of the Schreiber Transverse Mercator I
would be greatful for a copy.

> There is documentation available at the French Institut Géographique
> National, publication "Notes Techniques", NT/G 73.
> <http://www.ign.fr/affiche_rubrique.asp?rbr_id=1700&lng_id=FR#68096>
> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
>
> This publication gives the algorithms of two more Gauss-Laborde type
> projections, one with a "sphère equatoriale" and one with a "sphère
> bitangente".
> The "sphère de courbure" projection was or is still in use for Réunion.
> I have no idea where the other two Labordes were used for.
>
> More background information in:
> ENSG-IGN, Didier Bouteloup, Cours de Géodésie, Chapitre 3: §4.5.d
> "Représentation de Gauss-Laborde"
> <http://www.ensg.ign.fr/~bouteloup/www/wwwfad/site_fad/pdf/index_pdf.htm>
> <http://www.ensg.ign.fr/~bouteloup/www/wwwfad/site_fad/pdf/chap3.pdf>
>
> Example:
> International ellipsoid;
> lat=-21; lon=55.5;
> lat0=-21d7m; lon0=55d32m; x0=1.6e5; y0=5e4; k0=1;
> x,y Laborde type 1 = stmerc = 156534.17713, 62916.92507
>
>
>
> _______________________________________________
> Proj mailing list
> [hidden email]
> http://lists.maptools.org/mailman/listinfo/proj

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary

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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
In reply to this post by OvV_HN
On Monday 12 June 2006 5:47 am, Oscar van Vlijmen wrote:
> stmerc is a projection in the libproj4 package. There is no documentation.
> Mr. Evenden mentioned it on this discussion list on 2004-03-26.
> Recently I found a connection with a projection in real use, the
> Gauss-Laborde type "sphère de courbure" (probably meaning: the projection
> sphere has a radius equal to the radius of curvature at the point of
> origin).
> What is the connection? They are the same!

Looking at the NT/G 76, the tmerc projection described uses the isometric
latitude as the intermediate latitude while stmerc in libproj4 uses the Gauss
conformal latitude which is more complex in its conversion and has side
effects such as creating a new radius (see the pj_gauss documentation in the
libproj4 manual).

However, other complicating factors later in the 76 development prohibits me
from saying it is not the same as stmerc.  I will, however, safely say that
is it not equivalent to Gauss-Kruger that we all know and love. :-)

I think I will code 76 up and see how things do compare.  Thankfully, C99 has
complex arithmetic and math library.  I must add that NT/G series are quite
useful for developing code.

BTW, I am having a devil of a time accessing the French site and keep timing
out.  It took me nearly an hour to just access three of the documents and the
browser finally completely gave up on the last one.  All other net access
seems normal.

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary

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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

OvV_HN
>> stmerc is a projection in the libproj4 package. There is no documentation.
>> Mr. Evenden mentioned it on this discussion list on 2004-03-26.
>> Recently I found a connection with a projection in real use, the
>> Gauss-Laborde type "sphère de courbure" ...

Mr. Evenden wrote:

> Looking at the NT/G 76, the tmerc projection described uses the isometric
> latitude as the intermediate latitude while stmerc in libproj4 uses the Gauss
> conformal latitude which is more complex in its conversion and has side
> effects such as creating a new radius (see the pj_gauss documentation in the
> libproj4 manual).
...
> I think I will code 76 up and see how things do compare.  Thankfully, C99 has
> complex arithmetic and math library.  I must add that NT/G series are quite
> useful for developing code.

Going complex is not needed. Those few sums of complex sines can easily be
computed doing Re and Im separately in reals.

<http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
NTG 73 describes 3 types of Gauss-Laborde projections. One of them, the
"sphère de courbure", is equal to stmerc.
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.

<http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_76.pdf>
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.

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
<http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatnin
g/geodesi/Formelsaml%C3%ADng/Gauss_Conformal_Projection.pdf>
3) Finnish JHS
<http://www.jhs-suositukset.fi/intermin/hankkeet/jhs/home.nsf/files/JHS154/$
file/JHS154.pdf>
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.

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.





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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
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.
>
> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
> 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.
>
> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_76.pdf>
> 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
> <http://www.lantmateriet.se/upload/filer/kartor/geodesi_gps_och_detaljmatni
>n g/geodesi/Formelsaml%C3%ADng/Gauss_Conformal_Projection.pdf>
> 3) Finnish JHS
> <http://www.jhs-suositukset.fi/intermin/hankkeet/jhs/home.nsf/files/JHS154/
>$ file/JHS154.pdf>

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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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

OvV_HN
>> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
>> 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.

My interpretation of "sphère de courbure" was:
the projection sphere has a radius equal to the radius of curvature at the
point of origin. But how is this sphere positioned? Tangent at the origin
(lat0, lon0) perhaps?
The "équatoriale" case has something to do with the projection sphere
positioned on the equator. In the "bitangente" case probably the projection
sphere touches the ellipsoid twice, but where?
Schreiber knew; he developed several double projections!

>> 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.
I understand that among others, ESRI has been busy developing such a TM. So
there must be a professional need for it.

> BTW: what did you use to get the "exact" values?
My own stuff.
I had a very hard time finding solid code. Several people are guarding the
principles as a secret and are deliberately vague. Or they are trying to get
solid money from it by selling software or books.
But I persevered in trying to get the Dozier show on the road.
You (mr. Evenden) already found one error in his code, but this error has to
be corrected in 3 places. It's the error of the elliptic parameter m, which
has to be the elliptic modulus k in 3 cases (tmfd, gk, tmid).
It appeared that the Dozier code - the complex Newton iteration - was
useless in some regions, especially large lon-lon0 and low latitudes. First
I used somewhat better elliptic functions from Cernlib. But, to more effect,
I concocted another iteration scheme based on TOMS algorithm 365, a very
slow downhill walkaround method, yet very powerful.
I checked my results with an on-line calculator from professor Schuhr, based
on the Klotz algorithms.
<http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>
This calculator fails for difficult areas (very large lon-lon0, small lat).
So I can only 'proof' my results in the difficult areas by doing a complete
round-trip and getting nearly the original data back.
And yes, I do the lon-lon0=90 degrees too.

The Dozier article:
<http://www2.bren.ucsb.edu/~dozier/publications.htm>



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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Clifford J Mugnier
In reply to this post by Gerald I. Evenden-2




The "sphère de courbure" is merely the Gaussian Sphere evaluated at the
geodetic latitude of origin.

Melita Kennedy of ESRI informed me a couple years ago that they
successfully programmed Jeff Dozier's TM equations and have incorporated
that routine into one of the ESRI packages, I don't remember which one.

Clifford J. Mugnier C.P.,C.M.S.
National Director (2006-2008)
Photogrammetric Applications Division
AMERICAN SOCIETY FOR PHOTOGRAMMETRY AND REMOTE SENSING
and
Chief of Geodesy
Center for GeoInformatics
LOUISIANA STATE UNIVERSITY
Dept. of Civil Engineering
CEBA 3223A
Baton Rouge, LA  70810
Voice:              (225) 578-8536
Facsimile:          (225) 578-8652
-----------------------------------------------------------
http://www.ASPRS.org/resources/GRIDS
http://www.cee.lsu.edu/facultyStaff/mugnier/index.html
-----------------------------------------------------------


>> <http://www.ign.fr/telechargement/MPro/geodesie/CIRCE/NTG_73.pdf>
>> 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.

My interpretation of "sphère de courbure" was:
the projection sphere has a radius equal to the radius of curvature at the
point of origin. But how is this sphere positioned? Tangent at the origin
(lat0, lon0) perhaps?
The "équatoriale" case has something to do with the projection sphere
positioned on the equator. In the "bitangente" case probably the projection
sphere touches the ellipsoid twice, but where?
Schreiber knew; he developed several double projections!

>> 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.
I understand that among others, ESRI has been busy developing such a TM. So
there must be a professional need for it.

> BTW: what did you use to get the "exact" values?
My own stuff.
I had a very hard time finding solid code. Several people are guarding the
principles as a secret and are deliberately vague. Or they are trying to
get
solid money from it by selling software or books.
But I persevered in trying to get the Dozier show on the road.
You (mr. Evenden) already found one error in his code, but this error has
to
be corrected in 3 places. It's the error of the elliptic parameter m, which
has to be the elliptic modulus k in 3 cases (tmfd, gk, tmid).
It appeared that the Dozier code - the complex Newton iteration - was
useless in some regions, especially large lon-lon0 and low latitudes. First
I used somewhat better elliptic functions from Cernlib. But, to more
effect,
I concocted another iteration scheme based on TOMS algorithm 365, a very
slow downhill walkaround method, yet very powerful.
I checked my results with an on-line calculator from professor Schuhr,
based
on the Klotz algorithms.
<http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>
This calculator fails for difficult areas (very large lon-lon0, small lat).
So I can only 'proof' my results in the difficult areas by doing a complete
round-trip and getting nearly the original data back.
And yes, I do the lon-lon0=90 degrees too.

The Dozier article:
<http://www2.bren.ucsb.edu/~dozier/publications.htm>



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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
On Tuesday 13 June 2006 4:14 pm, Clifford J Mugnier wrote:
> The "sphère de courbure" is merely the Gaussian Sphere evaluated at the
> geodetic latitude of origin.
>
> Melita Kennedy of ESRI informed me a couple years ago that they
> successfully programmed Jeff Dozier's TM equations and have incorporated
> that routine into one of the ESRI packages, I don't remember which one.

I would only ask one question: was the code in the appendix of Dozier's
article without error?

I bashed my head against that stuff several years ago to no avail.  I even
converted the code into computer readable form two ways: by scanner and OCR
and brute force keying in.
        ...
--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary

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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
In reply to this post by OvV_HN
On Tuesday 13 June 2006 2:59 pm, Oscar van Vlijmen wrote:
        ...
> >> 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.
>
> I understand that among others, ESRI has been busy developing such a TM. So
> there must be a professional need for it.

It is an interesting academic problem but I fail to see any practical
application.  Scale factor and large scale distortion goes to hell rapidly
from the central meridian for any expanded Mercator aspect and thus
unsuitable for cadastral or grid systems.  Old style navigation that
justified Mercator navigations charts seems to be on the way out.  So other
than demonstrating loxodromes what is the use of continental size
Mercator---especially for the ellipsoid?

It seems peculiar that someone who has to watch expenditures would spend time
on this activity.  There must be an unknown, external driving dollar.

Reminds me of the book about men who watch goats.

> > BTW: what did you use to get the "exact" values?
>
> My own stuff.
> I had a very hard time finding solid code. Several people are guarding the
> principles as a secret and are deliberately vague. Or they are trying to
> get solid money from it by selling software or books.

You may be right, but I do not see any visible results yet.  But again, I
cannot see anyone in their right mind buying such stuff without there being a
strong NEED.

> But I persevered in trying to get the Dozier show on the road.
> You (mr. Evenden) already found one error in his code, but this error has
> to be corrected in 3 places. It's the error of the elliptic parameter m,
> which has to be the elliptic modulus k in 3 cases (tmfd, gk, tmid).
> It appeared that the Dozier code - the complex Newton iteration - was
> useless in some regions, especially large lon-lon0 and low latitudes. First
> I used somewhat better elliptic functions from Cernlib. But, to more
> effect, I concocted another iteration scheme based on TOMS algorithm 365, a
> very slow downhill walkaround method, yet very powerful.
> I checked my results with an on-line calculator from professor Schuhr,
> based on the Klotz algorithms.

I believe I saw a reference to the Klotz article the calculator page.  Is it
worthwhile?

> <http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>

Interesting site but the above url failed on my browsers and I had to back up
two levels and guess my way in.  At the moment I have not figured out enough
German to effectively work the calculator.  I have an easier time with
French, can slowly work through simple German and throw up my hands in defeat
with Polish.  :-(  And working in Grads is a bit of a nuisance.

An unrelated aside: most people are unaware that proj/lproj accepts radian
input---suffix the value with r (0.123r).  It might be an idea to add a g
suffix for grad.

> This calculator fails for difficult areas (very large lon-lon0, small lat).
> So I can only 'proof' my results in the difficult areas by doing a complete
> round-trip and getting nearly the original data back.
> And yes, I do the lon-lon0=90 degrees too.
>
> The Dozier article:
> <http://www2.bren.ucsb.edu/~dozier/publications.htm>

Dozier was nice enough to send me a copy but i got the feeling later that he
had lost interest in the project.

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

strebe
In reply to this post by OvV_HN

You might contact Dr. David E. Wallis. He devised a much simpler method than Dozier's. I've implemented it for the full-ellipsoid. You can see a plot of an earth-like ellipsoid here:

http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Mercator.GIF

The method works for arbitrary eccentricities. Contact me privately if you're interested. Since it is Dr. Wallis's invention, I'll put you in contact with him.

Regards,
-- daan Strebe


In a message dated 6/13/06 12:03:39, [hidden email] writes:


> BTW: what did you use to get the "exact" values?
My own stuff.
I had a very hard time finding solid code. Several people are guarding the
principles as a secret and are deliberately vague. Or they are trying to get
solid money from it by selling software or books.
But I persevered in trying to get the Dozier show on the road.
You (mr. Evenden) already found one error in his code, but this error has to
be corrected in 3 places. It's the error of the elliptic parameter m, which
has to be the elliptic modulus k in 3 cases (tmfd, gk, tmid).
It appeared that the Dozier code - the complex Newton iteration - was
useless in some regions, especially large lon-lon0 and low latitudes. First
I used somewhat better elliptic functions from Cernlib. But, to more effect,
I concocted another iteration scheme based on TOMS algorithm 365, a very
slow downhill walkaround method, yet very powerful.
I checked my results with an on-line calculator from professor Schuhr, based
on the Klotz algorithms.
<http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>
This calculator fails for difficult areas (very large lon-lon0, small lat).
So I can only 'proof' my results in the difficult areas by doing a complete
round-trip and getting nearly the original data back.
And yes, I do the lon-lon0=90 degrees too.




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Re: Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
On Wednesday 14 June 2006 1:12 am, [hidden email] wrote:

> You might contact Dr. David E. Wallis. He devised a much simpler method
> than Dozier's. I've implemented it for the full-ellipsoid. You can see a
> plot of an earth-like ellipsoid here:
>
> http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Mercat
>or. GIF
>
> The method works for arbitrary eccentricities. Contact me privately if
> you're interested. Since it is Dr. Wallis's invention, I'll put you in
> contact with him.

Dr. Wells has a web page relating to the projection:

http://www.wallisphd.com/mercator.htm

that a Google search on his name will return.  Found this site several years
ago during a previous discussion about tmerc.  To me, the web page appears
unchanged and the "Publication Pending" notice at the top is certainly taking
a long time.

I should post him a letter for further information and follow up with a phone
call if no response.

The description of the equation present does not make much sense unless what
looks like p in the tan term is not the p described as colatitude.  Again,
the first line says p,lambda is the colatitude and longitude yet the
description following the formula talks about a p for the Elliptic Integral
of the second kind.

I must be missing something.

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

strebe

Hm. I didn't know about that web page. Obviously it's wrong -- for some
reason "p" appears in several different roles. I tend to think that's
an error in conversion to a web page. (I see that the entire blurb is a
single graphic, not HTML mark-up.) Certainly he's been pedantic and
precise in all his communications with me.

The p/2 exponent should read (e/2), where e is the eccentricity.

Use some other variable (perhaps p') in place of p in "Then, the
complex variable tan (p/2) can be obtained..." and "...yields the
argument p..."

Regards,
-- daan Strebe


-----Original Message-----
From: Gerald I. Evenden <[hidden email]>
To: PROJ.4 and general Projections Discussions <[hidden email]>
Sent: Wed, 14 Jun 2006 11:42:40 -0400
Subject:  Re: [Proj] Re: Discovery: libproj4 stmerc = French
Gauss-Laborde   projection

   On Wednesday 14 June 2006 1:12 am, [hidden email] wrote:
> You might contact Dr. David E. Wallis. He devised a much simpler
method
> than Dozier's. I've implemented it for the full-ellipsoid. You can
see a
> plot of an earth-like ellipsoid here:
>
>
http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Merc
at
>or. GIF
>
> The method works for arbitrary eccentricities. Contact me privately if
> you're interested. Since it is Dr. Wallis's invention, I'll put you in
> contact with him.

Dr. Wells has a web page relating to the projection:

http://www.wallisphd.com/mercator.htm

that a Google search on his name will return.  Found this site several
years
ago during a previous discussion about tmerc.  To me, the web page
appears
unchanged and the "Publication Pending" notice at the top is certainly
taking
a long time.

I should post him a letter for further information and follow up with a
phone
call if no response.

The description of the equation present does not make much sense unless
what
looks like p in the tan term is not the p described as colatitude.  
Again,
the first line says p,lambda is the colatitude and longitude yet the
description following the formula talks about a p for the Elliptic
Integral
of the second kind.

I must be missing something.

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

OvV_HN
In reply to this post by strebe
From: Strebe-aol.com

> You might contact Dr. David E. Wallis. He devised a much simpler method than
> Dozier's. I've implemented it for the full-ellipsoid. You can see a plot of an
> earth-like ellipsoid here:
>http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Mercator.G
IF

Thanks for the nice picture!



A word of caution for all TM methods:
it is very difficult to get any accuracy at the edges, i.e. for very large
lon-lon0 at small values of lat, and very near the poles. Let's not
exaggerate our claims!

By using for instance an approximation like the Finnish Julkisen Hallinnon
Suositus or the French IGN uses, even the mathematical singularity at the
poles (lat=90) seems to vanish (Wallis claim).

It apparently seems not too difficult to devise a reasonable TM
approximation for the entire (hemi)sphere, good enough for large-scale
mapping. But to get high accuracy at the edges is very difficult.


From: gerald.evenden-verizon.net

>> I checked my results with an on-line calculator from professor Schuhr,
>> based on the Klotz algorithms.
>> <http://gauss.fb1.fh-frankfurt.de/>
> And working in Grads is a bit of a nuisance.

The German 'Grad' is the English 'degree'.
English 'grad' was in German 'Neugrad', now 'Gon'.


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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
In reply to this post by OvV_HN
On Tuesday 13 June 2006 2:59 pm, Oscar van Vlijmen wrote:
        ...
> I checked my results with an on-line calculator from professor Schuhr,
> based on the Klotz algorithms.
> <http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>
> This calculator fails for difficult areas (very large lon-lon0, small lat).
> So I can only 'proof' my results in the difficult areas by doing a complete
> round-trip and getting nearly the original data back.

I finally cased out usage of the above site and agree that within the limits
of 6 degrees from the central meridian lproj agrees to about .1mm.  However, I
did note that the above site's northing fails near the pole.  This is
dependent upon the accuracy of the meridianal distance algorithm and where
lproj's should be good to about 14 digits or about .1 nm.

Given the labor of hand keying test values on an online site, the test was not
very rigorous.

I might have been able to push meridianal distance to IEEE 64 bit float limits
but my interest in the project was began to wane.

With the exception of northing, lproj seems to agree to the .1mm of the site's
display precision within the 3.5 degree range of UTM.  This range is certain
the practical limits of TM.
--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Martin Vermeer
In reply to this post by strebe
On Wed, 2006-06-14 at 13:50 -0400, [hidden email] wrote:
> Hm. I didn't know about that web page. Obviously it's wrong -- for some
> reason "p" appears in several different roles. I tend to think that's
> an error in conversion to a web page. (I see that the entire blurb is a
> single graphic, not HTML mark-up.) Certainly he's been pedantic and
> precise in all his communications with me.
>
> The p/2 exponent should read (e/2), where e is the eccentricity.

Yes, I agree.

> Use some other variable (perhaps p') in place of p in "Then, the
> complex variable tan (p/2) can be obtained..." and "...yields the
> argument p..."


Actually the argument p is simply the (ellipsoidal) co-latitude
90d - phi.

The common expression in u and v corresponds to exp(psi), where psi is
the _isometric latitude_, i.e., essentially the "northing" in a
traditional (non-transverse) Mercator map plane.

Isometric latitude and longitude (psi, lambda) together as (x,y)
co-ordinates in a plane define a conformal mapping from the curved
Earth's surface. Using (psi, lambda) directly as rectangular
co-ordinates produces classical Mercator. Using

u + iv = exp(psi + i * lambda)

i.e., polar co-ordinates, produces the stereographic projection. This is
very much what Dr Wallis's formula looks like. Apparently for him it is
only a trick leading somewhere... but then I also get lost.

Regards Martin V

PS you may want to look at

http://users.tkk.fi/~mvermeer/geom.pdf

pp 99-100 and around p. 90. Sorry it's in Fenno-ugrian formulese...


> Regards,
> -- daan Strebe
>
>
> -----Original Message-----
> From: Gerald I. Evenden <[hidden email]>
> To: PROJ.4 and general Projections Discussions <[hidden email]>
> Sent: Wed, 14 Jun 2006 11:42:40 -0400
> Subject:  Re: [Proj] Re: Discovery: libproj4 stmerc = French
> Gauss-Laborde   projection
>
>    On Wednesday 14 June 2006 1:12 am, [hidden email] wrote:
> > You might contact Dr. David E. Wallis. He devised a much simpler
> method
> > than Dozier's. I've implemented it for the full-ellipsoid. You can
> see a
> > plot of an earth-like ellipsoid here:
> >
> >
> http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Merc
> at
> >or. GIF
> >
> > The method works for arbitrary eccentricities. Contact me privately if
> > you're interested. Since it is Dr. Wallis's invention, I'll put you in
> > contact with him.
>
> Dr. Wells has a web page relating to the projection:
>
> http://www.wallisphd.com/mercator.htm
>
> that a Google search on his name will return.  Found this site several
> years
> ago during a previous discussion about tmerc.  To me, the web page
> appears
> unchanged and the "Publication Pending" notice at the top is certainly
> taking
> a long time.
>
> I should post him a letter for further information and follow up with a
> phone
> call if no response.
>
> The description of the equation present does not make much sense unless
> what
> looks like p in the tan term is not the p described as colatitude.  
> Again,
> the first line says p,lambda is the colatitude and longitude yet the
> description following the formula talks about a p for the Elliptic
> Integral
> of the second kind.
>
> I must be missing something.
>
> --
> Jerry and the low-riders: Daisy Mae and Joshua
> "Cogito cogito ergo cogito sum"
>    Ambrose Bierce, The Devil's Dictionary
> _______________________________________________
> Proj mailing list
> [hidden email]
> http://lists.maptools.org/mailman/listinfo/proj
>
>
> ________________________________________________________________________
> Check out AOL.com today. Breaking news, video search, pictures, email
> and IM. All on demand. Always Free.
>
> _______________________________________________
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> [hidden email]
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
In reply to this post by Gerald I. Evenden-2
On Wednesday 14 June 2006 9:38 pm, Gerald I. Evenden wrote:
        ...
> This is dependent upon the accuracy of the meridianal distance algorithm
> and where lproj's should be good to about 14 digits or about .1 nm.

        Duh! Sorry, I should have said .1 um (micrometer) rather than nano.

--
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"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Curt, WE7U
On Thu, 15 Jun 2006, Gerald I. Evenden wrote:

> On Wednesday 14 June 2006 9:38 pm, Gerald I. Evenden wrote:
> ...
> > This is dependent upon the accuracy of the meridianal distance algorithm
> > and where lproj's should be good to about 14 digits or about .1 nm.
>
> Duh! Sorry, I should have said .1 um (micrometer) rather than nano.

Question:  Isn't "nm" also an abbreviation for nautical mile?  A bit
of a difference between nautical mile and nanometer.

--
Curt, WE7U.   APRS Client Comparisons: http://www.eskimo.com/~archer
"Lotto:    A tax on people who are bad at math." -- unknown
"Windows:  Microsoft's tax on computer illiterates." -- WE7U
"The world DOES revolve around me:  I picked the coordinate system!"
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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

OvV_HN
In reply to this post by Gerald I. Evenden-2
> From: "Gerald I. Evenden" <gerald.evenden-verizon.net>

>> I checked my results with an on-line calculator from professor Schuhr,
>> based on the Klotz algorithms.
>> <http://gauss.fb1.fh-frankfurt.de/cgi-bin/cgi_gk>

> I finally cased out usage of the above site and agree that within the limits
> of 6 degrees from the central meridian lproj agrees to about .1mm.
...
> With the exception of northing, lproj seems to agree to the .1mm of the site's
> display precision within the 3.5 degree range of UTM.  This range is certain
> the practical limits of TM.

I ran a script, comparing the Dozier transverse Mercator with complex Newton
iteration, and tmerc.
Everything under IEEE-754 math; differences smaller than 1e-6 have probably
lost some significance.

cnewton iteration
wgs84 ellipsoid
lat0=0; lon0=0; x0=0; y0=0; k0=1;

Searching for the maximum differences in x and y per longitude, stepping 1
degree in latitude.

lon = 0.5...[0.5]...6...[1]...12...[3]...21
   lat = 0...[1]...89

lon-lon0 (deg), lat (deg) / max diff (m)
0.5, 0 / dxmax 2.4e-9
0.5, 84 / dymax 1.5e-8
1, 1 / dxmax 5.2e-8
1, 88 / dymax 3.9e-8
1.5, 0 / dxmax 4.1e-7
1.5, 89 / dymax 3.4e-8
2, 0 / dxmax 1.81e-6
2, 16 / dymax 5.9e-8
2.5, 0 / dxmax 5.85e-6
2.5, 16 / dymax 2.3e-7
3, 0 / dxmax 1.55e-5
3, 15 / dymax 7.3e-7
3.5, 0 / dxmax 3.61e-5
3.5, 15 / dymax 1.93e-6
4, 0 / dxmax 7.63e-5
4, 15 / dymax 4.55e-6
4.5, 0 / dxmax 0.000150
4.5, 15 / dymax 9.80e-6
5, 0 / dxmax 0.000277
5, 15 / dymax 0.0000197
5.5, 0 / dxmax 0.000489
5.5, 15 / dymax 0.0000373
6, 0 / dxmax 0.000831
6, 15 / dymax 0.0000676
7, 0 / dxmax 0.00218
7, 15 / dymax 0.000198
8, 0 / dxmax 0.00516
8, 14 / dymax 0.000518
9, 0 / dxmax 0.0113
9, 14 / dymax 0.00123
10, 0 / dxmax 0.0231
10, 14 / dymax 0.00273
11, 0 / dxmax 0.045
11, 14 / dymax 0.00571
12, 0 / dxmax 0.0836
12, 14 / dymax 0.0113
15, 0 / dxmax 0.433
15, 14 / dymax 0.0695
18, 0 / dxmax 1.758
18, 13 / dymax 0.326
21, 0 / dxmax 5.99
21, 13 / dymax 1.26

So, till a longitude difference of 6 degrees tmerc has better than
millimeter accuracy. Beyond 9 degrees no longer centimeter accuracy.

Additionally: libproj's mdist (meridional distance) is very accurate, about
machine precision, compared to a formula based on the elliptical integral.
But I can't recommend mdist for eccentricities larger than about 0.9.



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Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
On Thursday 15 June 2006 2:16 pm, Oscar van Vlijmen wrote:
        ...
> Additionally: libproj's mdist (meridional distance) is very accurate, about
> machine precision, compared to a formula based on the elliptical integral.
> But I can't recommend mdist for eccentricities larger than about 0.9.

I don't know at what point we would be spinning off the planet.  Should I put
a caveat about this limit?  ;-)

--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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Re: Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

Gerald I. Evenden-2
In reply to this post by strebe
On Wednesday 14 June 2006 1:12 am, [hidden email] wrote:

> You might contact Dr. David E. Wallis. He devised a much simpler method
> than Dozier's. I've implemented it for the full-ellipsoid. You can see a
> plot of an earth-like ellipsoid here:
>
> http://mapthematics.com/Projection%20Images/Cylindrical/Transverse%20Mercat
>or. GIF
>
> The method works for arbitrary eccentricities. Contact me privately if
> you're interested. Since it is Dr. Wallis's invention, I'll put you in
> contact with him.

I wrote to the address on the web site but letter was returned undeliverable.  
I suspect that the web page is several years old and not maintained.

Not to beat a dead horse of several years ago, I have stared at the above gif
and it still bothers me and it does not seem real.  The cusps at the ends of
the equator seem unnatural.  Also, why does both of the other procedures that
we have looked at all contain discontinuities at the limits---most commonly,
they require isometric latitude which fails at 90 degrees.  What magic twist
allows Wallis to come up with the above map when all others want to extend to
infinity?  Is it truly a normal transverse mercator where the scale factor is
1. along the central meridian?  Has that been checked?  Sorry, I am still a
skeptic until I see the math and a functional program that can demonstrate
the conformal properties of the projection.

The cusps at the ends of the equator sure look like violations of conformality
to me.

As previously noted, the French TM has been added to libproj4 and the Dozier
procedure has also been pretty well conquered and will be add to
libproj4---probably as dtmerc.  Neither of these routines will do |lat|=90
nor |lon|=90.
--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary
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