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    Re: Obtaining Azimuths. was: Re: Burdwood's Tables
    From: John Cole
    Date: 2007 Oct 14, 06:21 -0700

    While looking over my two pages of mind numbing manual interpolation I
    found an error in the last step and decided to organize the process on
    an Excel sheet. This led to Az of 285.16 for Example 1 and 284.65 for
    Example 2, Close to George's results of 285.1 and 284.7 respectively.
    
    John Cole
    
    On Oct 11, 9:24 pm, John Cole  wrote:
    > 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
    > 285.0
    >
    > John Cole
    >
    > On Oct 11, 5:16 am, "George Huxtable" 
    > wrote:
    >
    > > 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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