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

[Strebe at aol.com]

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.

> 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.
> 
> Note that the cusp (poorly displayed in the previously mention gif url)
> appears to be the beginning of the multiple root solution.  I say poorly
> displayed as the gradation of the equitorial parallel should *smoothly* 
> begin
> a swing to the north OR (importantly) to the south. 
> 
The cusp is poorly observed, not poorly displayed. The illustration is exact 
to sub-pixel resolution and it matches Lee's illustration precisely despite 
using a different method of calculation. It seems you do not know a continuously 
differentiable curve when you see one.

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

To that end, please feel free to contine the discussion on the "Complex 
Transverse Mercator" thread.

Regards,
-- daan Strebe


In a message dated 6/29/06 08:45:00, gerald.evenden at verizon.net writes:

> I think the basic idea presented by Dozier sounds feasible but his 
> execution
> falls short of the goal.
> 
> Although the the method of computing elliptic integral of the second kind 
> with
> complex argument may be OK it has troubles with large arguments when
> evaluating the Jacobian Zeta function.  The second, and I believe the more
> serious problem, 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 and searching through the net and all leads me to the
> conclusion that we are getting into deep water when dealing with the complex
> 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.
> 
> Note that the cusp (poorly displayed in the previously mention gif url)
> appears to be the beginning of the multiple root solution.  I say poorly
> displayed as the gradation of the equitorial parallel should *smoothly* 
> begin
> a swing to the north OR (importantly) to the south.  Selecting the north or
> south root becomes a practical problem for a projection program and is a
> problem with any method dealing with the comprehensive TM projection.
> 
> Because I have no training and no experience in working with complex 
> variable
> problems and have failed to find any practical material related to alternate
> methods to compute elliptic integrals with complex arguments and, more
> importantly, a Newton-Raphson routine for determining roots of complex
> functions, I have decide to suspend again any activity on the Dozier method.
> 
> One has to know when to throw in the towel.  ;-)
> --
> Jerry and the low-riders: Daisy Mae and Joshua
> "Cogito cogito ergo cogito sum"
>    Ambrose Bierce, The Devil's Dictionary
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