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__.
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. |