# [Proj] Re: Dozier's TM method---my summary

Gerald I. Evenden gerald.evenden at verizon.net
Tue Jul 4 23:10:38 EDT 2006

```On Thursday 29 June 2006 3:30 pm, Strebe at aol.com wrote:
> In a message dated 6/29/06 08:45:00, gerald.evenden at verizon.net writes:
> (Full text of message at end.)
>
> > ...is the Newton-Raphson method employed.  He has expanded the
> > basic real function and applied it to a complex variable.  I am not sure
> > that
> > this is appropriate...
>
> I haven't looked at Dozier in depth, but in general there is nothing less
> appropriate about Newton-Raphson when applied to complex variables than
> there is when applied to real-valued variables. It's quite easy to get
> yourself into trouble when finding roots even of real-valued functions.
> While the function in question has problematic regions, that has nothing to
> do with the fact that it is complex-valued.

Isn't that what I just said???  At this point I am picking on Dozier, not you
nor anyone else.

> > plane.  Also, it looks like we are also dealing with multiple roots,
> > especially when longitude exceeds a certain value   (suggested to be
> > (pi/2)*(1-k))---a factor not addressed in Dozier's solution
>
> I do not understand why you think the fact of multiple roots supports your
> notion of the intractability of the solution.

Where did I say the solution did not exist nor the plot was not basically
correct---just poorly done.

... material previously commented on

> You have remarked previously that the projection is "not intuitive". To
> you, of course. Not necessarily to the reading audience. You didn't like

Let us stop for a moment on this "intuitive issue."  Plain old Mercator goes
to infinity in the NS directions for both the sphere and elliptical case.
When we flip the cylinder over on its side the projection goes to infinity in
the east-west direction (as one might logically expect) for the spherical
case.  SO, is it not intuitive that that the elliptical case would behave in
a similar manner?  That is all I am saying.  I am perfectly willing to accept
that for some weird magic of complex analysis that it defies intuition and is
finite in all directions.  I have accepted this finiteness on faith for
several years, but being a good atheist, I must allways view it with
skepticism until it is appropriately demonstrated.

> the cusp; you thought it indicated the projection wasn't really conformal.
> Once the cusp was demonstrated (by means of a 30-year old peer-reviewed
> journal article) to be an attribute of the projection, you decided you

I had the impression you got the plot from Wallis.  Was he peer reviewed?  If
so, what was the 30 year old publication.

> didn't like how it looked in an image I supplied, even though that image
> precisely matches the one on the peer-reviewed article. It is curious that
> you believe it appropriate to cast public aspersion based on your own,
> flawed notions rather than objective facts. If that is how you talk
> yourself out of a project then I don't suppose there is much anyone can do
> about it, since reason clearly has nothing to do with it.
>
> To those interested in the full-spheroid transverse Mercator, I urge you
> not to be skeptical of its existence or attainability based on Mr.
> Evenden's comments. There is nothing controversial about either. The
> mathematics has been published in peer-reviewed journals, confirmed any
> number of times by people who understand the mathematics, and expressed by
> at least three different calculational methods (whatever method Lee used;
> Dozier; and Wallis) in at least five implementations that I know of. While
> there is treacherous calculational territory to traverse, that is true of
> many projections. As always, you must understand the domain and choose
> numerical techniques appropriate to it.

Stating it another way, my surrender was to the Dozier article that while the
method sounds interesting the execution falls short.  The code had several
errors and some methodology is in question.  I have spent an additional week
rooting out a complex Bulirsch routine from the old Bell lab site and spent
several days relating Jacobi's form to Legendre in order to test against
Abramowitz tables and GSL software for at least real arguments---they now
agree.  But Dozier's code does not agree to better than the third or forth
place (I am not saying Dozier's code is wrong yet).

And I still have no handle on a replacement for Newton-Raphson.

There's the old saying: "I'm from Missouri.  Show me."  Has someone have a
reasonable reference list on this subject (other than Dozier and Wallis).  I
am aware of Lee but I have not seen some of the other names mentioned without
details.

> To that end, please feel free to contine the discussion on the "Complex
> Transverse Mercator" thread.
>
> Regards,
> -- daan Strebe

Lastly, I am not a grad student dependent upon the graces of the almighty
professor and thus must speak in muted and humble tones in his presence.

Rather than complaining about my skepticism, I'd appreciate a good pointer or
hints on complex root finding.  :-)
--
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
Ambrose Bierce, The Devil's Dictionary

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