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    Re: Finding The Symmedian
    From: George Huxtable
    Date: 2010 Dec 23, 19:32 -0000

    Andrew Nikitin's posting, earlier today, repeats the algorithm provided on
    the back pages of the Nautical Almanac, as Peter Hakel has reminded us.
    
    But, as I have pointed out more than once before, what is described in the
    Almanac is an iterative procedure, and I take it that what Andrew has
    described is just one round of that iteration (though I'm unable to follow
    the Reverse Polish logic of his procedure).
    
    The next step is to look at the change in position that the procedure has
    calculated, and if that exceeds some decided preset amount (the Almanac
    suggests 20 miles) to recalculate intercepts and azimuths from that new
    position, and go through another round of the procedure. To be useful, his
    program should include that whole reiteration. Otherwise, it's only
    applicable if the AP is known sufficiently well, beforehand.
    
    ====================================
    
    I wrote, on 9 December, in the thread "A simple three-body fix puzzle"
    
    "The Almanac doesn't explain well, and Frank hasn't either, that this is
    just one step in an iteration, which starts from a guessed initial
    position, such as the DR position. The next step is to test whether the
    resulting change in position is outside some preset limit, and the Almanac
    suggests 20 miles. Depending on how close the DR position was, a single
    iteration may well suffice. It depends.
    
    If another iteration is called for (and this is the bit that isn't
    explained well) then all the azimuths and intercepts have to be
    recalculated, using the new values of assumed long and lat, and the
    summations repeated using those new values, and these are used to provide a
    better estimate still. Convergence is usually rapid."
    
    George.
    
    contact George Huxtable, at  george@hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    ----- Original Message -----
    From: "Andrew Nikitin" 
    To: 
    Sent: Thursday, December 23, 2010 2:50 PM
    Subject: [NavList] Re: Finding The Symmedian
    
    
    Regarding finding most probable position.
    
    The equations involved are so simple that could be solved on a
    calculator.  Admittedly, use of calculator is "cheating", but
    much less so than a laptop.  I would also guess that calculator
    does not take nearly as much space on a charting table as
    laptop.
    
    Also, when finding MPP digitaly, any number of LOPs can be
    used, not just 3.
    
    Here is a reminder on how to do it.
    
    If all LOPs are represented as azimuth and offset vectors
    relative to same origin (such as AP), the corresponding
    equations can be written out without even plotting them
    first.  For example, for LOP with given AZ and D the equation
    would be x*sin(AZ)+y*cos(AZ)=D
    
    Each sight contributes one equation. For 3 sights we have an
    overdefined system of 3 equations with 2 variables (x y).  These
    equations can be solved using least square method (which exactly
    minimizes sum of square distances from point (x,y) to each of
    the lines).  Substituting our equation coefficients into formula
    and making long things short we get the following solution:
    
    A=sum(sin(AZi)^2)
    B=sum(sin(AZi)*cos(AZi))
    C=sum(cos(AZi)^2)
    D=sum(sin(AZi)*Di)
    E=sum(cos(AZi)*Di)
    DET=A*C-B*B
    x=(D*C-E*B)/DET
    y=(E*C-D*B)/DET
    
    (Here AZi,Di are azimuth vectors and offsets of LOPs, and
    resulting x and y is an offset vector relative to same origin as
    LOPs)
    
    Solving it by punching keys is tedious, but this is what
    programmable calculators are for. The direct implementation of
    the above formulas should not be too difficult.
    
    Here is, for example, what I wrote for HP35s. It is not
    especially complicated program and it wasn't hard to write.
    Program for other calculators will be very similar in
    complexity.
    
    (XEQ L002 initializes routine, than user enters AZi <ENTER> Di
    <R/S> And when all pairs are entered, XEQ L003 calculates dx and
    dy)
    
    L001 LBL L
    L002 GTO L004
    L003 GTO L034
    L004 0
    L005 STO A
    L006 STO B
    L007 STO D
    L008 STO E
    L009 STO C
    L010 R/S
    L011 STO F
    L012 X<>Y
    L013 STO H
    L014 COS
    L015 STO G
    L016 *
    L017 STO+ E
    L018 RCL H
    L019 SIN
    L020 STO H
    L021 RCL* F
    L022 STO+ D
    L023 RCL H
    L024 X^2
    L025 STO+ A
    L026 RCL H
    L027 RCL* G
    L028 STO+ B
    L029 1
    L030 STO+ C
    L031 RCL C
    L032 R/S
    L033 GTO L011
    L034 RCL C
    L035 RCL- A
    L036 STO H
    L037 RCL* A
    L038 RCL B
    L039 X^2
    L040 -
    L041 1/x
    L042 STO G
    L043 RCL H
    L044 RCL* D
    L045 RCL E
    L046 RCL* B
    L047 -
    L048 *
    L049 RCL H
    L050 RCL* E
    L051 RCL D
    L052 RCL* B
    L053 -
    L054 RCL* G
    L055 RTN
    
    
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