# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: LOP's without DR (so-called "Douwes's problem")
From: Antoine Couëtte
Date: 2015 Feb 22, 22:31 -0800

Hello again to all,

Just a quick note to (tentatively) clarify the kind of solutions which have been published to solve our "LOP's without DR" problem.

In the Real World, the specific points to be solved are twofold:

- The Observers are generally moving, and successive heights are taken from different points on Earth or at Sea. And

- The heights measurements are not "exact" or "perfect" in the mathematical sense because of unavoidable errors (instrument errors, dip and refraction errors, and/or other errors). In addition to these errors the mathematical modeling of the Observers' movements may also be in error (differential DR error).

The published solutions fall into 2 different families:

- ONE PURELY ANALYTICAL SOLUTION using only 100% Calculus and no Geometry at all. This is the "French air Force Colonel's solution" on a spherical Earth for steady observers (and maybe even for steadily moving Observers ?). This method has some limited practical interest since - to the best of my memories - it works only for "perfect" height measurements and bodies with zero parallax (stars case). Nonetheless it keeps its own GREAT MERIT of exactly solving the "LOP's without DR" problem through only Calculus.

- 3D GEOMETRICAL SOLUTIONS SOLVED THROUGH CALCULUS. If the starting points of these Solutions seem to be quite different and even far away from one another, they all eventually fall onto the very same family of equations to be solved through Calculus. If the Observers were steady and if the heights measurements were perfect, such methods also immediately yield the right Observers' positions on a spherical Earth for bodies with zero parallax (star cases). Late Professor Georges Bodenez's published method falls into this category. In order to cope with the Real World (moving observers, various inaccuracies mentionned hereabove, finite distance bodies with appreciable parallax seen from an ellipsoidal Earth), these methods remain ITERATIVE BY NATURE. And in particular since the Parallax correction is Latitude dependent on an Ellipsoid, there is no way out of iterations. How would it be possible to adequately compute such Parallax corrections for finite distance bodies if no approximate DR positions are even and ever available ? Iterations remain the only recourse here.

*******

Again, rather than giving a full description of these methods with their respective merits, I am much more interested in knowing whether somebody published a solution to our "LOP's without DR" Problem before Professor Georges Bodenez published a solution in France in 1976.

*******

Antoine M. "Kermit" Couëtte

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