Welcome to the NavList Message Boards.

NavList:

A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

Compose Your Message

Message:αβγ
Message:abc
Add Images & Files
    or...
       
    Reply
    Re: Sight Reduction by the Cosine Haversine Method
    From: George Huxtable
    Date: 2004 Oct 3, 15:05 +0100

    Chuck Taylor has taken some of the mystery out of haversines and the
    cosine-haversine method, and shown how such calculations were made.
    
    I think there's a bit more to be add, if Chuck doesn't mind.
    
    ====================
    
    He wrote-
    >Haversines are merely a vehicle for simplifying the
    >computations.  While sines and cosines range from -1
    >to + 1, haversines range only from 0 to + 1, and the
    >haversine of a negative angle is the same as the
    >haversine of the absolute value of that angle.
    
    That's all correct, but Chuck has omitted the important reason WHY it was
    necessary to avoid negative values. The reason was the USE OF LOGS.
    
    Unless a navigator was keen to do 4- or 5-figure long-multiplication and
    long-division (and who would be?), before the days of calculators and
    computers such calculations had to be done by logs. Logs were necessary and
    useful, but had a great drawback; that for negative numbers, the logarithm
    is meaningless. There was a trick to make get such log calculations to work
    with negative numbers, but it never found favour with navigators. Instead,
    their calculation formulae were rewritten in a form so that when logs were
    taken, the numbers were always positive.
    
    First, a new function, the "versine", (1 - cos), was introduced, in
    addition to the usual sin and cos; clearly, this was always positive,
    varying between 0 and +2. Later, it became clear that a function which
    never exceeded 1 would be more useful still, so the versine was simply
    halved, becoming the haversine (hence its name). Then the altitude formula
    was reconstructed to use the haversine and avoid negative quantities.
    
    Things have changed. In these days of pocket-calculators and computers,
    there's no need to involve logs or haversines, and everything becomes much
    simpler. But Chuck's description is useful, and interesting, in showing how
    the old navigators had to do it, and you can do it their way if you wish.
    
    To compute an intercept, you need lat and dec, and the separation in
    longitude, (or HA, which Chuck abbreviates as t) and then require the alt,
    to compare with the corrected alt from your sextant.
    
    ===================
    
    Direct method (can't be used with logs).
    
    From the basic cosine formula for a spherical triangle, then
    
    sin alt = sin lat sin dec + cos lat cos dec cos HA
    
    With a calculator, that's easy to work out. For this method, you have to
    remember to give lat and dec their appropriate signs, + for North, - for
    South. The sign of HA (i.e. whether it's East or West) doesn't matter.
    
    ==================
    
    Cosine-haversine method (using logs).
    
    If you're interested in how the haversine calculation works out in the
    method Chuck describes, the expression above can be manipulated into-
    
    hav ZA = hav (lat ~ dec) + sin lat sin dec hav HA
    
    ZA is the zenith angle, or (90 - alt)
    
    Lat and dec are now always positive quantities, 0 to 90 degrees, whatever
    the hemisphere.
    
    (lat ~ dec) is the amount of North-South angular difference between them,
    taking account of which hemisphere each is in, therefore in the range 0 to
    + 180 degrees.
    
    Instead of sin lat cos dec hav HA in the expression above, we can
    substitute hav X; we don't need to bother what X actually is, as will be
    seen.
    
    So hav ZA = hav (lat ~ dec) + hav X
    
    and hav X = sin lat sin dec hav HA
    
    Then taking logs,
    
    log hav X = log sin lat + log sin dec + log hav HA    These logs always
    turn out now to be of positive quantities, so that's OK.
    
    Chuck's procedure adds those logs to get log hav X.
    
    Most haversine tables (but not all) list the angle (0 to 180 deg), then its
    log hav, and then its hav (as "nat", or natural, haversine). In my modern
    edition of Norie's (1970), this table, to 5 decimal places, occupies over
    100 pages.
    
    So, knowing the log hav, you search down that column for the value of log
    hav X that you have just calculated. Then alongside it in the adjacent
    column, is the value of the haversine, hav X. You don't need to bother to
    find what the corresponding value of angle X actually is.
    
    So now we have hav X, to which we must add hav (lat ~ dec), and the result
    is hav ZA.
    
    Then take out the angle ZA, corresponding to this value of hav, from the
    table, subtract it from 90 degrees, and you have the required calculated
    alt, to compare with the sextant and obtain the intercept.
    
    =====================
    
    The method for deriving azimuth, from its cosine, quoted by Chuck, has two
    problems.
    
    One was the difficulty that for azimuths near due East, say, angles the
    same amount North and South of East, such as 80degrees and 100 degrees,
    have the same cosine, so the expression for cos az can't distinguish
    between them. Similarly for azimuths near West. A special procedure was
    quoted to distinguish which was which, which added quite a bit of
    complexity.
    
    By the way, if observations of the Sun were made between the Autumn and
    Spring equinox, there's no need to make such a test, because during the
    Winter months the Sun must always be in the Southern quadrants from the
    Northern hemisphere (and vice versa). Similarly for stars with a South
    declination observed from the Northern hemisphere.
    
    The other difficulty is that cos az changes hardly at all for angles near
    to East and West, so in that situation, the azimuth is predicted only
    approximately and is very sensitive to small errors in calculation.
    
    Unless one is forced to use logs, both these difficulties are avoided by
    using a different expression which derives az from tan az and is better in
    every way. This alternative has been referred to earlier on Nav-L and I
    won't discuss it further unless anyone asks.
    
    George.
    
    ================================================================
    contact George Huxtable by email at george---.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    ================================================================
    
    
    

       
    Reply
    Browse Files

    Drop Files

    NavList

    What is NavList?

    Join NavList

    Name:
    (please, no nicknames or handles)
    Email:
    Do you want to receive all group messages by email?
    Yes No

    You can also join by posting. Your first on-topic post automatically makes you a member.

    Posting Code

    Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.
    Email:

    Email Settings

    Posting Code:

    Custom Index

    Subject:
    Author:
    Start date: (yyyymm dd)
    End date: (yyyymm dd)

    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site