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Jeff

I have played with the provide formula and it appears one can enter the
course in cardinal coordinates (decimal degrees) directly.  No conversion
from Cn to C seems to be needed.

It would also appear that S and E must be negative.

I used several scenarios to compare the formula to the trig/Bowditch
meridional parts method and polar-to-rectangular calculator method.

In mid latitudes they are identical for all practical (light-displacement
boat) purposes.  The formula vs. meridional parts vs. calculator
polar-to-rectangular do show differences near the equator over a relatively
short distance.

Given:

Course 160 true
Distance 350 nm
L1 5d 55' North
l1 80d 36' East

Formula           Bowditch/Trig    Calculator P-R
L2  00d 26!1 N    L2  00d 26!1 N   L2 00d 26!1 N
l2  82d 35!9 E    l2  82d 35!1 E   l2 82d 35!9 E

Bill

>> I actually succeeded in using Paul Hirose's suggestion to "rotate" the
>> triangle, substitute values, and use complements of values (90 - A).
>> But I failed miserably in trying to simplify the formulae by
>> substituting the complementary trig functions.  If I knew what I was
>> doing (or received several more hints) I'd bet there was a much simpler
>> formula hiding in there somewhere.
>
> If understood your initial question, it was:
>
> If I start out at a known position, and sail a constant course for a given
> number of miles (nautical), where will I wind up?  You desire your new
> location to be given as latitude and longitude.
>
> Susan P Howell's book, "Practical Celestial Navigation" gives three methods
> under the chapter heading, "Mercator Sailing."
> 1.  Formula
> 2.  Simpler trig formula and Bowditch meridional-parts table
> 3.  Totally tabular with Bowditch traverse table
>
> You asked for a formula, so I will try my best to relate it via text.
>
> L1 = starting latitude
> L2 = finishing latitude
> l1 = starting longitude
> l2 = finishing longitude
> C = Course
> D = Distance
> ln = natural log
>
> NOTE:  East and South are negative
>
> L2 = [(D cos C)/ 60] + L1
>
> l2 = l1 - tan C {180 [ln tan (45 + .5 L2) - ln tan (45 + .5 L1)] / pi}
>
> In the reverse of the this formula (given L1, l1, L2 and l2, find course and
> distance) Howell notes that the you must inspect the direction between the
> two points to be sure course is measured eastward from north.  C is labeled
> north or south depending on direction between L1 and L2. C is labeled east
> or west depending on direction between l1 and l2.
>
> Then the C derived is corrected to actual course (Cn) via the following:
> Cn = NE
> Cn = 360 - NW
> Cn = 180 - SE
> Cn = 180 + SW
>
> She does not state if course in the above formulas need to be adjusted from
> Cn to C.  I will leave that to you to discover.
>
>
> I am bad at converting formulas to ASCII, and worse at typing, so will scan
> the page and email to you at your request.
>
> Since I am already using a calculator to do the above, for a distance of
> less than 500 nm in my neck of the woods, I use a derivation of mid-latitude
> sailings and the polar-to-rectangular feature on my TI-30Xa (which is very
> intuitive).  I can enter the course in cardinal degrees and tenths and
> obtain my corrections to L1 and l1 in nautical miles.  Latitude corrections
> can be taken as minutes with no conversion.  The main tricks here are to
> remember x and y distance answers are swapped as we are not in using trig
> coordinates (y will become longitude, and x latitude); and longitude
> distance must be converted from nautical miles to degrees and minutes
> longitude.  This is basically as simple of adding L1 to L2, dividing that by
> 2, and taking its cosine.  Invert the cosine (1/X key), then multiply that
> by the nm distance for minutes longitude (convert to degrees/minutes if
> necessary) to add or subtract from l1.
>
> If interested, I have the above polar-to-rectangular (as well as
> rectangular-to-polar) written up with illustrations in Microsoft word, and
> would be glad to send them to you at your request.  It sounds painful at
> first, but after you do it a few times, it is almost automatic.
>
> Hope the above is of some use.
>
> Bill
>
> billyrem42@earthlink. net

   

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