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    Re: a go at Silicon Sea:Leg 81
    From: Michael Wescott
    Date: 2001 Oct 19, 1:40 PM

    Peter Fogg wrote:
    
    >>  3) What is the DR position at the end of this time?
    >  N10d52'.6 W131d 59'.5
    
    >> 5) What is the TC & Distance from the DR to our Hawaii
    > landfall point at 19d 30.0'N 154d 45.0'W
    
    > Approx. 1400nm at 292d TC (I'll have to find a way to work this out
    > accurately - any clues?)
    
    Here are 3 ways.
    
    1) Use a Great Circle calculation. Treat the destination as star.
       LHA is difference in Longitude. Declination == Destination Latitude
       Distance is (90d - Hc) * 60.  TC = Z. The problem with this method
       is that's not how you'll sail. The TC changes as you progress and
       must be continually recalculated to sail a Great Circle course.
    
    2) Plane Sailing. Treat the problem as a simple triangle. The North-South
       distance is simply the latitude difference (in degrees) times 60 to
       give nautical miles. The East-West difference is the difference in
       Longitude (times 60) and times the cosine of the Mid-Latitude.
       Unfortunately the mid-Latitude is not so easily calculated. But a
       reasonable approximation is to split the difference, and maybe shade
       it a little closer to the pole. Pythagorean theorem gives us the
       distance and simple trigonometry for the course. This is the easiest
       to calculate with a simple calculator (with sin, cos, sqrt and inverse
       functions).
    
    3) Meridional parts is a more accurate variation on the theme of Plane
       Sailing. Instead of factoring the change of scale into the Longitude
       difference, here it is done to the Latitude. Each latitude is converted
       into meridional parts, the difference of the MP's is taken as the
       N-S side of the triangle and the Longitude difference (aka depature)
       is the E-W side. The course is computed from this triangle. The distance
       is computed by dividing the Latitude difference (in minutes) by the
       cosine of TC (ignoring signs). This will tend to be a bit more accurate
       except where the courses are almost due East or West. The calculations
       are bit more complex, but can be easily programed or approximated by
       a simple series.
    
    The Meridional parts method usually gives the best answer with respect
    to the Silicon Sea series.
    
    For much better and more detailed explanation of all of this see chapter
    24 of Bowditch:
    
            http://pollux.nss.nima.mil/NAV_PUBS/APN/Chapt-24.pdf
    
    and for details on Meridional parts calculation, see the explanation of
    table 6.
    
            http://pollux.nss.nima.mil/NAV_PUBS/APN/Tables/TblExpl.pdf
    
    
    
    Using Meridional Parts (#3) on this problem:
    
    d.Lat = 19d 30.0'N - 10d52.6'N = 8d 37.4' = 517.4 nm.
    d.lLo = 154d 45.0'W - 131d 59'.5W =  22d 45.5' = 1365.5'
    d.MP = MP(19d 30.0'N) - MP(10d52.6'N) = 1185.58 - 652.21 = 533.37
    
    TC = 270 + arctan(533.37/1365.5) = 270 + 21.3 = 291.3
    Dist = 517.4/sin(21.3) = 1422.1
    
    
    
    
    Using Plane Sailing (#2):
    
    D.Lat =  517.4 nm
    departure = d.Lo * cos((19.5+10.8)/2) = 1365.5 * 0.965245 = 1318.0
    dist = sqrt(1318.0^2 + 517.4^2) = 1416.0
    TC = 270 + arctan(517.4/1318.0) = 291.4
    
    
    
    
    For me, I have trouble holding course within 2.5d for any length of time
    let alone worry about a .1d difference. And the distance calculation
    difference is 6 miles out of 1400 or 0.4%.
    
    
    
    
    --
            Mike Wescott
            Wescott_Mike{at}EMC.COM
    

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