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    Re: Position from crossing two circles : was [NAV-L] Reality check
    From: George Huxtable
    Date: 2006 Jun 8, 23:21 +0100

    Michael Dorl wrote-
    | Bear with me...
    |
    | I'm not thinking of any kind of graphical solution but rather the
    spherical
    | trig behind it. Consider two observations A and B.  Call the zeniths
    ZA and
    | ZB and the great circle between them AB.  Now think of a great
    circle
    | passing through ZA offset from AB by some angle X. We can write
    equations
    | for coordinates of points on the equal altitude circle for A as a
    function
    | of X, ZA, and the observed altitude of A.  Those equations represent
    places
    | we could have been when we made observation A. Now since the
    equations give
    | us a starting position, can't we extend the starting point by the
    vector
    | representing the distance we traveled between observing A and B and
    write
    | equations for the ending point as a function of X, ZA, and the
    distance
    | sailed vector? I understand  this requires that we make some
    assumptions
    | about the shape of the Earth. Also we end up with whatever errors
    occur
    | when translating speed - time - bearing
    | data to great circle data but these errors will arise in any
    treatment of
    | this problem. Ok, so now we have equations for our ending position
    as a
    | function of X, ZA, the altitude of A, and the distance sailed
    vector.  Now
    | we can write equations for the observed altitude
    | of B from our ending position as functions of X, ZA, the altitude of
    A, the
    | distance sailed  vector and ZB.  Solve these equations for X knowing
    the
    | observed altitude of B.  There will in general be two solutions on
    opposite
    | sides of AB.  Knowing X, we can compute our
    | starting and ending positions.
    
    ==============================
    
    It's at times like this that we really need that Nav-l blackboard.
    I've been puzzling over those words, trying to picture what Michael is
    putting across. We have a great circle AB between the zenith points ZA
    and ZB. And we have another great circle which also passes  through ZA
    but is offset from AB by some angle X. It's the relation between those
    great circles that I haven't taken in. In what way is the angle X
    measured? What does it represent?
    
    Michael adds- I understand  this requires that we make some
    assumptions
    | about the shape of the Earth. Also we end up with whatever errors
    occur
    | when translating speed - time - bearing
    
    Let's be clear about it. The distortion of a position circle, when
    every point on it is shifted through the same course and distance,
    into a non-circle has nothing to do with the details of the shape of
    the Earth. It occurs for a spherical Earth, which is the normal
    assumption in all these discussions. Nor does it relate to differences
    between great-circle and rhumb-line displacements. The distortion
    occurs in the same way when a position circle is shifted due North, in
    which case the shift is both great-circle and also rhumb-line.
    
    I have little doubt that it would be possible to create some
    expression that defined the resultant distorted circle, but it's
    unlikely to be any simple function; more like a numerical
    construction.  It would be centred on the first GP, after that had
    been displaced through the course and distance of the shift, but
    instead of being a circle on a polar diagram, defined by radius = 90 -
    alt, it would be more like
    radius = (90 - alt ) - some correction term * sin 2 (az - course).
    That would vary in the right sort of way.
    
    And then some form of simultaneous equation would be required to find
    the intersections with the true circle that surrounds the GP of the
    body of the second observation, either analytically or numerically.
    Doesn't seem a simple business, to me.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    

       
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