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

Strebe at aol.com Strebe at aol.com
Thu Jun 29 15:30:24 EDT 2006

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
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

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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