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

Gerald I. Evenden gerald.evenden at verizon.net
Thu Jun 29 11:41:20 EDT 2006

```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
```