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    Re: Obtaining Azimuths. was: Re: Burdwood's Tables
    From: John Cole
    Date: 2007 Oct 11, 19:24 -0700

    There are no warnings in H.O. 171 about errors or instructions as to
    interpolation. There is a single worked example given from which one
    deduces that interpolation has been used however the resulting azimuth
    was given to the nearest whole degree only.
    I worked George's examples two ways. First a quick and dirty run
    through rounding the LHA to 3 hr 38 min the Dec to 55 deg 30 min and
    the Alt to 61 deg 30 min and interpolating for LHA in the table.  The
    answer was Az  285 to the nearest whole degree.
    The second time through I did not round anything  but interpolated
    carefully throughout including the last step for Az.  Unfortunately
    after all that work I got the same answer for both examples, namely
    John Cole
    On Oct 11, 5:16 am, "George Huxtable" 
    > I have altered the threadname because we are now comparing various sources
    > for obtaining azimuths.
    > Thanks to John Cole for his late recantation about message-sizes, and for
    > providing-
    > | Another historic table for finding azimuths is found in H.O. No. 171 "Line
    > | of
    > | Position Tables (for working sight of heavenly body for line of position
    > by
    > | the cosine-haversine formula, Marcq Saint Hilaire Method)"  US Navy Bureau
    > | of
    > | Navigation 1915. Table V the Finding of the Azimuths, page attached.
    > |
    > | The table is entered with the dec (across the column headings) and hour
    > | angle (in hr and min, down the column) to find a tabulated number. Then
    > the
    > | tabulated number is located again in the dec column whose heading is
    > closest
    > | to Hc and the Az is read off in the hour angle column and its direction
    > | determined by the usual rules.
    > |
    > | The rest of the tables in HO 171 are altitude corrections and log sines,
    > log
    > | cosines, and log and nat haversines.
    > =========================
    > That table is the same, in principle, as the one provided by Bennett. They
    > are both tabulations of a number which corresponds to cos X sin Y, where X
    > corresponds to the column heading, and Y the row heading. Bennett writes the
    > result as a 3-figure digit, the nearest that corresponds to 1000 cos X sinY.
    > In HO 171, it's tabulated instead as log( cos X sin Y), to 5 decimal places,
    > which uses more space. [Note that those are "navigator's logs", to which 10
    > has been added to turn negative numbers into positive ones, not exactly the
    > logs you will find in school log-tables or on a calculator.]
    > Both tables use the formula sin Az = sin LHA cos Dec / cos Alt.
    > We can rewrite that as          cos Alt sin Az = cos Dec sin LHA.
    > Conveniently, both sides of this expression have the same form.
    > So first find a number (or perhaps its log) that corresponds to cos Dec Sin
    > LHA. Then look for that same number (or same log) in the Alt column, and it
    > will be in the row that corresponds to Az.
    > Rounding errors in HO171 will in general be somewhat less than Bennett's
    > because the columns are at half-degree rather than whole-degree intervals,
    > though the rows are the same, and because there's no rounding (to speak of)
    > of the tabulated number.
    > It would be of some interest to me if John Cole would kindly extract
    > azimuths from HO171 for the contrived extreme examples I used when testing
    > Bennett's tables, as follows-
    > Example 1. Dec 55d 31'N, LHA 54d 29', alt 61d 29'. The true result for
    > Azimuth should be 285.1 degrees.
    > Example 2. Dec 55d 29'N, LHA 54d 31', alt 61 d31'. The true result should be
    > 284.7 degrees.
    > Even if HO171 values for those examples are less erroneous, it's likely that
    > at other combinations, some significant errors may occur. I wonder if it
    > carries any warnings about where it should not be relied on. I wonder, also,
    > whether any advice is given about interpolating between whole-number values.
    > George.
    > contact George Huxtable at geo...---.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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