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    Re: Obtaining Azimuths. was: Re: Burdwood's Tables
    From: George Huxtable
    Date: 2007 Oct 11, 11:16 +0100

    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 george---.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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