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    Re: Two-body fix caveat
    From: George Huxtable
    Date: 2009 Nov 4, 23:48 -0000

    I'm a bit puzzled by Peter Hakel's "illustration" in the two pages attached 
    to [10414], although the more-recent mathematical talk is quite beyond my 
    The discussion seems to be about whether various mathematical techniques 
    produce a "fix". I would not describe the position circles in that example, 
    which are osculating rather than intersecting, as producing a "fix". Both 
    are tangential to a North-South line passing through 20� West. Although the 
    point on the equator at 20� West may be the only point that actually 
    satisfies the two altitudes, other positions, to its North and South, come 
    so infinitesimally close to meeting that requirement that the result is 
    really a "line-fix". It's an example of the degenerate situation, of two 
    bodies with the same azimuth, that navigators know to avoid. In reality, 
    they would choose another body, or wait a bit, in time.
    In those circumstances, it's no surprise to me that difficulties arise in a 
    mathematical solution. A real navigator, working from a simple plot, would 
    realise what was going wrong immediately.
    On another matter altogether, Peter has experienced some problems in 
    quadratic fitting with Excel. So have I, using Excel 2000, and in doing the 
    same job, fitting a quadratic for longitude-around-noon. Excel will 
    cheerfully plot a quadratic fit to such a set of points, and apparently a 
    good fit too, and if asked to, will display the resulting coefficients, as 
    numbers printed on top of the fitted graph. And the only way I've found to 
    extract those coefficients, to use for further Excel calculation, is to read 
    them off that graph, and then type them back in again. Frustrating, but 
    acceptable. But, having done just that, and then used those numbers to 
    regenerate the quadratic curve once again, for checking, the resuting plot 
    can differ significantly from the quadratic that Excel had plotted before. 
    The reason is that in  displaying those parameters, they are truncated by 
    rounding-off in such a way as that significant digits have been lost. I've 
    got round it by renumbering the divisions along the x-axis, so its numerical 
    values increase by 10, or perhaps 100, accordingly.
    If Peter can offer any insight into how to get at those coefficients 
    directly, or how to alter their rounding-off, it would be helpful.
    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. 
    NavList message boards: www.fer3.com/arc
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