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Sailings The relationships can be laid out as two triangles. The sides of the top triangle are Distance, difference in latitude (dLat) and Departure. Departure is the longitude distance in miles. The lower triangle relates Departure to the difference in longitude (dLong.) The angle labelled mLat stands for Mean Latitude.
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From the top triangle:- dLat = Distance x Cos (Course) Eq 1 Departure = Distance x Sin (Course) Eq 2 From the lower triangle:- dLong = Departure/ Cos(Mean Latitude) Eq 3 Tabulated values are found in Nautical Tables as “Traverse Tables.” However the simplicity of the formulae are ideal for calculators.
Example Initial Position 45° N 30° W Course 045° T Distance 100’
Using the first formula (Eq 1): dLat = 100 x Cos(45°) dLat = 70’.7 = 1° 10’.7 Final Latitude 45° + 1° 10’.7 = 46° 10’.7 N
From the second formula (Eq 2); Departure = 100 x Sin (045°) = 70.71
From the third formula (Eq 3) dLong = 70.71/ Cos ( 45° + 70.7/ 2) dLong = 101’.04 = 1° 41’.0 Therefore the final longitude = 30° + 1° 41’.0 = 31° 41’.0 W
For a Noon calculation, dLat is the difference between the calculated and observed latitudes. The "Course" becomes the direction of the Position Line.
Distance = dLat/ Cos(P/L Dirn) Departure = Distance x Sin(P/L Dirn) dLong = Departure/ Cos(Mean Latitude) | |||||||||
Plane Sailing is adequate for distances up to about 60’. If the example is worked using Mercator Sailing, there is a difference of 0’.3 in the final longitude.
Mercator Sailing allow for the oblate shape of the Earth. (A squashed sphere.)
The formulae are dLat = Dist x Cos(Course) As for Plane Sailing - Eq 1 and dLong = Tan(Course) x DMP Eq 4
DMP stands for Difference in Meridian Parts. Meridian Parts are the distance in nautical miles from the equator to the required latitude. These are tabulated in Nautical Tables.
Example Initial Position 45° N 30° W Course 045° T Distance 100’
Using the first formula (Eq 1): dLat = 100 x Cos(45°) dLat = 70’.7 = 1° 10’.7 N because the track is northerly Final Latitude 45° + 1° 10’.7 = 46° 10’.7 N
Meridian Parts for 45° 3013.38 for 46° 10’.7 3114.08 Difference in Meridian Parts 100.7
From the second formula (Eq 4); dLong = Tan (045°) x 100.7 = 100’.7 = 1° 40’.7
Original Longitude 30° 00’.0 W dLong 1° 40’.7 West because the course is westerly Final Longitude 31° 40’.7 W
The drawback with Mercator Sailing is the need to refer to tables. If you use a programmable calculator then this is the formula to calculate Meridian Parts. (The infinite series of terms in Bowditch et al is simply an expansion of this.)
A x Ln(Tan( 45° + Lat/ 2)/ ((1 + e x Sin(Lat))/ (1 – e x Sin(Lat)) ^ (e/ 2) For WGS84; A = 3437.74677 and e = 0.08182
Note that most nautical tables are based on the Clarke 1880 spheroid that uses a different compression to WGS84. The difference is small but noticeable (0.27 at 45°.) For Clarke 1880 use e = 0.08248. | |||||||||
Next Section Celestial Navigation Calculations
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