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    Azimuth determination (ad nauseum)
    From: Peter Fogg
    Date: 2007 Oct 6, 21:06 +1000

    While some of us have heard all this time and again, its been a couple of years now since George dragged up last his same tired old objection to the Azimuth Tables in The Complete On-Board Celestial Navigator, so we'll go though it all again for the benefit of those who haven't heard it all before.

    The book devotes quite a bit of attention to the derivation of azimuth, beginning with:

    General advice on azimuth determination

    As long as the DR is relatively close to the position, intercepts remain short (remember that the tabular sight reduction method proposed in the book is based on a single DR, not an assumed position to nearest whole degrees).  In this case the need for accuracy with azimuths is not, generally speaking, great.  Additionally, given the limitations on plotting with great precision while underway on a small boat, azimuths accurate to within a couple of degrees are adequate for the purpose.

    Indeed, a corrected compass bearing should give such a reasonably good idea of true bearing, or at least serve as a check for whatever other method may be used; indicating gross error if such should occur.  Or vice versa.  Redundancy is good on boats.

    While on the subject of estimation of azimuth, the prediction and identification tables of the book can be used, although since latitude and LHA to the nearest 10 degrees are used: "the accuracy of azimuths found by this method may not be very high".

    For plotting purposes the book proposes quite a few alternatives:


    Azimuth Tables

    Apparently this simple and quick tabular method of deriving azimuth was popular with sailing ships in the nineteenth century. This is hardly surprising since the method: 

    "… involves one table with different points of entry. No interpolation is required, and it is one of the simplest techniques for finding azimuth with an accuracy of one or two degrees."

    Sounds great, doesn't it? They are great too; they were popular for good reason (and still are, with people like me who actually use them). Is there a catch? Well, its a good idea to understand their limitations.

    If I remember the story correctly, these tables are based on the sine formula, and so tend to become unreliable as the azimuth approaches the prime vertical – due east and west. Of course the remedy in such cases is obvious – beware.  In practice, in such a case the initial problem can be choosing which of two similar values to adopt.  There is a procedure to follow for this, but since it compromises the 'quick and simple' appeal of the tables, I find it altogether simpler to give these tables a miss with azimuths around prime vertical.

    Is that it? Not quite. The tables are entered with values of whole degrees. George has rather cleverly discovered that if a carefully contrived combination of half degree values are used, significant error can be induced. And is still dining out, from time to time, on this.

    A statistical study has been carried out to analyse and quantify this potential problem. The results have never been challenged.  When 121,677,000 random combinations of possible values were tested, an azimuth correct to within 2 degrees was generated in 98.8% of the cases. "Very large errors i.e. greater than ten degrees occur in only 0.005% of the sample. An error in excess of ten degrees up to a maximum of less than 18 degrees is extremely rare."  More detail is available at:


    under 'Azimuth Tables'.

    The common sense approach is to beware of entering the tables with a combination of half degrees, especially when the bearing approaches east and west.  Once the navigator is aware of the extremely rare potential for significant error in such a combination of circumstances its easy to avoid, via use then of an alternative method. As Frank says; its a trivial problem, not a major one, except in George's mind. Although we have gone over all this again and again and again, generally to deafening silence from George, every so often George likes to climb back up onto this worn soapbox and moan piously about how dangerous such potential error could be for the poor navigator. You're pushing piffle, George. You really should get a life.

    Additionally, in a few succinct words the book explicitly warns the user of the above, concluding: "When in doubt, use the Weir diagrams" which are conveniently located on the page following the azimuth tables.


    Weir Diagrams

    Much like, I should imagine, generations of students of navigation, I have always thought of them as weird diagrams.

    I've heard that they were popular with the Royal Navy, which used larger coloured versions than the book, that is limited to line drawings that fit onto the pages.  They work fine, it seems (am unaware of any criticism).  Because they constitute a graphical solution, the accuracy of the derived azimuth is limited by the skill of the user (this also affects the accuracy of the plotted LOPs and then fix, with the pencil and paper plots proposed by the book). I guess that they are also, generally, limited to being accurate to within a degree or two. Maybe less.

    Is this good enough?  Sure it is.  It is unrealistic to expect great accuracy from celestial observations from the deck of a small boat underway. What are proposed by the book is relatively quick and simple methods that are entirely appropriate for that environment.

    But hey, you want precision?  Drag out that calculator …


    Azimuth via Formulas

    Three are proposed in the book:

    The tan formula, which George tells us he approves of.

    The cos formula, which I've been given to understand suffers when asked to produce azimuths close to due north and south, and:

    The sin formula, which has the aforementioned inbuilt lack of reliability with azimuths close to east and west.


    Is that enough about the derivation of azimuth from one slim volume?  I would have thought so.  Given all this choice I would have thought disingenuous was expressing George's obsession with the Azimuth Tables in the mildest terms possible.

    But wait, there's more:



    There's a formula for this too (and other stuff) although the Azimuth Tables can also be used to derive amplitude. They're handy tables, those Azimuth Tables.  I wouldn't leave home without them.


    A challenge for George

    If you want to show us what a clever chap you really are George, why don't you take your favourite tan formula and produce from it a set of tables that will derive azimuth at least as simply and easily as some creative person has done with the sin formula.  Of course we will expect it to work perfectly in all circumstances.  I should imagine that nothing less would be good enough for your own high standards, either.

    Do that George and we will be impressed.  You will have managed to actually create something of value, rather than just carping in an increasingly monotonous fashion from the sidelines about imperfections (and in particular a highly contrived error) within hugely useful tools that have been regularly used in the past by real sailing ships over extended periods and are again proposed for use, quite justifiably, albeit with their warts, for the benefit of us all.


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