[Proj] Re: Discovery: libproj4 stmerc = French Gauss-Laborde projection

[Strebe at aol.com]

 gerald.evenden at verizon.net writes:
(Original e-mail in full at the end.)

> 
> The cusps at the ends of the equator seem unnatural. 
> 
> ...

> The cusps at the ends of the equator sure look like violations of 
> conformality
> to me.
> 
> 
Do the cusps on the August epicycloidal or the Eisenlohr seem unnatural? If 
so, do you believe those projections are not really conformal? If not, then 
what do you think the difference is?

The position of the cusps is directly related to the eccentricity -- in fact, 
the relationship is surprisingly simple:

Longitude of cusp = (pi/2) * (1 - e) radians

The spherical form is infinite because the cusp is at the pole... which 
projects to infinity. The greater the eccentricity, the closer to the prime 
meridian starts the cusps, and the stubbier the map.


> Also, why does both of the other procedures that
> we have looked at all contain discontinuities at the limits
> 
> 
Why is 1 - sqrt (1 - k) when k approaches zero ill-conditioned when computed 
as is, but marvelously well-behaved when computed as a series? The question is 
not the mathematics of the projection. We're talking about the same 
projection, whether Dozier, Kruger, or Wallis. The question is only how the projection 
is formulated. Wallis's formulation avoids poor conditioning. They all express 
the same underlying mathematics.


> And a remaining note: this projection has not been published in an any
> cartographic journal and has not been subject to normal peer review nor the
> review of subsequent readers.
> 
> 
I think you mean to say that Wallis' formulation has not been published. 
Obviously the projection has, by any number of people over nearly two centuries.


> What magic twist
> allows Wallis to come up with the above map when all others want to extend 
> to
> infinity?
> 
> 
You are misundertanding the problem. The others don't "want" to extend to 
infinity; their accuracy simply degrades into uselessness.


> Is it truly a normal transverse mercator where the scale factor is
> 1. along the central meridian? 
> 
> 
Yes.


> Has that been checked? 
> 
> 
Yes, empirically, and yes, analytically.


> Sorry, I am still a
> skeptic until I see the math and a functional program that can demonstrate
> the conformal properties of the projection.
> 
> 
You might avail yourself of a copy of L.P. Lee's monograph, "Conformal 
Projections Based on Elliptic Functions", Cartographica, Monograph Number 16, 1976. 
Quoting verbatim from p. 97:

"The positive y-axis represents part of the equator, extending from lambda = 
0 to lambda = (pi/2)*(1-k)... At this point the equator changes smoothly from 
a straight line to a curve... The projection of the entire spheroid is shown 
in Fig. 46, again using the eccentricity of the International (Hayford) 
Spheroid. It can be seen that the entire spheroid is represented withing the finite 
area without singular points..."

If you will take your attention to page 99, you will see a rendition of the 
(only true) transverse Mercator projection for the ellipsoid, generated by 
means of Thompson's 1945 formulation, succeeding across the entire ellipsoid 
because Lee (or someone) took the time to reformulate the problematic regions to 
make them suitable for calculation. The diagram is indistinguishable from the 
one I posted online and drew using Wallis's formulation.

Dozier seems to follow the formulation of Thompson, more or less, which is 
probably why it runs into trouble at the extremes and is certainly why it's more 
complicated than necessary. Wallis solves the boundary condition in a 
different way. It's not without its problems; you need to find a root which, for 
arbitrary eccentricities, defies any formulaic seed value. Nonetheless that can be 
solved efficiently in most cases and merely solved in all cases. There are 
one or two places where one must be careful of numeric techniques, like many 
other projections.

Regards,
-- daan Strebe


In a message dated 6/26/06 13:22:06, gerald.evenden at verizon.net writes:



> And a remaining note: this projection has not been published in an any
> cartographic journal and has not been subject to normal peer review nor the
> review of subsequent readers.
> 
> 


> On Wednesday 14 June 2006 1:12 am, Strebe at aol.com 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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