A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Andrés Ruiz
Date: 2006 Jun 8, 08:48 +0200
Geoge, the method is not impossible for n observations or running fix.
Mike, here you have the math for a fix from two circles of position (COP)
1. Position from two circles of equal altitude
The equation of the plane containing a COP in rectangular coordinates is: ax+by+cz-p = 0
For the two bodies you have two equations, two planes intersect in a line.
The two possible solutions for the observer’s position, P and P’, are the intersections of that line with the unit sphere x2+y2+z2 = 1
the math, (in C++):
2. Position from n circles of equal altitude
Here the problem is there are a lot of crossings between the circles. Metcalf & Metcalf, (On the overdetermined celestial fix - Refer to the Bibliography section at the link below), developed a method based on Lagrange Least-Squares minimization of the equation:
S ( Sin Ho – [ sin Dec sin Lat + cos Dec cos Lat cos(GHA+Lon) ]2 )
The result is the MPP(Lat, Lon) for n circles of position. No initial position is needed. Also support a running fix.
What is the C++ application you refer for calculate and plot the COP?, where can I found it?
Enviado el: miércoles, 07 de junio de 2006 13:39
Asunto: Re: [NAV-L] Position from crossing two circles : was [NAV-L] Reality check
At 06:10 AM 6/7/2006, George Huxtable wrote:
>I have written a program in bastard-Basic which runs on my 1980s Casio
>programmable calculator (FX 730P or FX 795P), and if anyone is
>interested would be happy to send it or post it up. It would be simple
>to adapt it to another machine. It takes the 6 quantities, dec, GHA,
>and altitude for each of two bodies, and returns two possible
>positions in terms of lat and long, for the user to choose the
>appropriate one. It does not require a DR or AP, and provides an exact
>result without going through an iteration process.
>It's not original, in that versions of the method have been described
>previously beforehand. For example, in an article by George Bennett in
>the journal "Navigation" (which is, I think, the American one) Issue
>no. 4, vol 26, winter 1979/80, titled " General conventions and
>solutions- their use in celestial navigation", and to the book
>"Practical navigation with your calculator", by Gerry Keys, (Stanford
>maritime, 1984), section 11.12. The method has also been described in
>"The K-Z position solution for the double sight", in European Journal
>of Navigation, vol.1 no, 3, December 2003, pages 43-49, but that
>article was bedevilled by printing errors that render it more-or-less
>unintelligible, which were corrected in a later issue. Not to mention
>several serious errors and misunderstandings by the author, which have
>never been acknowldged or corrected in that journal.
Do any of these sources spell out the math in detail? I've searched in
vain for a complete algorithm so a long time ago, I sat down and worked out
the math. One of the tricky things is determining what quadrant angles lie
in when doing a inverse trig function. I have a c++ windows application
which will find all the equal altitude circle intersections for a set of
observations. It also can plot the equal altitude circles on a world map.