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    Re: Bygrave formula accuracy on 10 inch slide rule
    From: Hanno Ix
    Date: 2009 Jul 8, 10:05 -0700
    Sir:

    This must be a most interesting analysis, and I wish you could post the results in a table or graph displaying the AZ/LAT- errors over an octant or what ever form you prefer.

    You know, when on sea you care more about your specific situation than an overall statistic no matter how good in general. So, such info would be most desirable.

    Regards



    --- On Tue, 7/7/09, glapook---.net <glapook{at}PACBELL.NET> wrote:

    From: glapook---.net <glapook{at}PACBELL.NET>
    Subject: [NavList 9019] Re: Bygrave formula accuracy on 10 inch slide rule
    To: "NavList" <NavList@fer3.com>
    Date: Tuesday, July 7, 2009, 10:06 PM


    The rms of 1.8 minutes using the Bygrave method is much better than
    the 4.7' error level you found using the standard sine-cosine formulas
    and confirms what everyone has written about the accuracy of the
    Bygrave. As you point out, the longer scale on a Bygrave would result
    in an even higher level of accuracy. Since you found 1.8 with just a
    ten inch standard slide rule the Byrave, which has a scale 7.85 times
    longer than a ten inch rule, would produce much better accuracy and
    certainly no worse accuracy than the ten inch rule. So the claim of
    one or two minute accuracy with a Bygrave appears to be validated and
    is consistent with my own tests on my recreation of the Bygrave.

    gl

    On Jul 6, 3:19 pm, Paul Hirose <cfuhb-ac...---.net> wrote:
    > My computer simulation of the Bygrave sight reduction formulas, worked
    > on a 10 inch slide rule, had altitude accuracy of 1.8' and azimuth
    > accuracy of 2.0'. Those are the square roots of the mean squared errors.
    >
    > In a run of 500,000 random sight reduction problems, 95% of the
    > solutions were correct within plus or minus 3.7' in altitude, and 95%
    > were within plus or minus 4.2' in azimuth.
    >
    > The maximum altitude error seen by the program was 10.4'. Worst cases
    > always seem to occur around 40° - 50° altitude. Note that the Bygrave
    > solution reads altitude on the tangent scale, which is most compressed
    > at 45°.
    >
    > The maximum azimuth error seen by the program was 28.0'. Worst cases
    > occur at high altitudes.
    >
    > I was suspicious of the accuracies reported by the test program. They
    > seemed too good, so I worked six random problems (generated by the
    > program) by hand on a 10 inch rule. Altitude errors (minutes) were +1.1,
    > -5.2, 0.0, +1.1, -2.3, -.6. Azimuth errors were -.8, +.1, +.3, -1.3,
    > +1.1, -1.4. These results suggest the program's modeling of slide rule
    > errors is realistic.
    >
    > To operate the slide rule I wore reading glasses but did not use my
    > hands free magnifying glass, though it would have helped a good deal.
    >
    > My program generates each sight reduction problem from a random azimuth
    > and altitude, the latter being weighted so the simulated stars tend to
    > have constant density everywhere in the sky instead of packing closer
    > with increasing altitude. Altitudes less than 5° or greater than 80° are
    > rejected.
    >
    > A random observer latitude between 0° and 70° is generated in similar
    > fashion.
    >
    > Each azimuth, altitude, and latitude combination is converted to LHA and
    > declination. The sight reduction module converts these values back to
    > azimuth and altitude, injecting a random error in each slide rule
    > operation, then compares results to the correct values.
    >
    > Slide rule error is assumed to be .1% RMS per multiplication or division
    > (which involves two settings and one reading). The Bygrave azimuth
    > formula requires *three* settings and one reading, so for that
    > calculation I increase the error accordingly.
    >
    > I have modeled the Bygrave formulas on a standard slide rule, but not
    > the Bygrave rule itself. Its error should be in inverse proportion to
    > its scale length relative to a 10 inch rule.
    >
    > --
    > I filter out messages with attachments or HTML.



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