Formulas and Procedures – Navigation.
Great Circles.
1. Great circle distance;
Cos Distance = Cos D’long Cos Lat A Cos Lat B ± Sin Lat A Sin Lat B
2. Vertex position;
Find initial course;
A = Tan Lat A = named opposite to Lat A unless D’long
Tan D’long is between 090˚ and 270˚
B = Tan Lat B = named same as Lat B
Sin D’long
C = Sum of A and B (named whichever is the greater)
Tan Azimuth = 1
C x Cos Lat A
Azimuth named N or S (as for C) θ˚ E or W (as for D’long)
Find Latitude of vertex and D’long between Longitude A and vertex using Napier’s Rules;
Sin Mid Part = Cos Opposite Parts
Or
Sin Mid Part = Tan Adjacent Parts
3. Distance off;
1. Find position of vertex
2. Find D’long between vertex and longitude of island / hazard.
3. Find latitude of course line at that longitude using Napier’s Rules
4. Find D’lat between course line and island / hazard = distance off.
4. Fuel problem;
1. Find distance to steam to burn fuel;
Distance = Fuel to burn x 24 hours
Daily consumption
2. Find longitude of vertex (Napier’s Rules)
3. Find remaining distance to steam (to burn fuel);
Distance = D’long
Cos M. Lat
4. Find remaining distance to steam to destination (great circle)
Cos distance = Cos D’long Cos Lat X Cos Lat B ± Sin lat X sin Lat B
(Lat X = Limiting Latitude)
5. Find total distance by adding the three parts;
Great circle + parallel sail + great circle.
Merpass;
1. Find ITP position (using calculator or scale drawing)
D’long = Departure
2. Find Easterly Hour Angle (EHA);
GHA (hour) =
Incr (m + s) =
GHA (obs) =
Longitude (ITP) = (-ve West, +ve East)
LHA (ITP) =
EHA (360˚ - LHA) =
3. Find Ship D’long in one hour;
D’Lat and departure found using calculator or scale drawing, then
4. Find Apparent Rate of Change(RoC);
Sun Hour Angle (15˚) ± Ship D’long ( - West, + East)
5. Find time to Merpass;
Time to Merpass = EHA
RoC
6. Find time of Merpass;
Initial time (GMT) =
Time to merpass = (+)
Merpass time =
Most Probable Position;
1. Lay out information given about stars in table form;
Time Star Azimuth Intercept Run
0515 A 075˚ 4’ away 2.0*
*Run is found by applying time star is taken to required time asked for:– general rule is if time of star is 0515 and time asked for is 0530, then apply run forward from DR position along course line for 15 minutes (distance = 15 ÷ 60 x ship’s speed)
2. Draw diagram;
3. From diagram, measure D’lat and departure, and then find D’long;
5. Apply D’Lat and D’Long to find MPP.
Sight;
1. Find approx GMT at ship, and apply this to chronometer reading to get actual GMT.
2. Find LHA (add longitude East, subtract longitude west) and declination.
3. Find True Altitude;
Sextant Alt =
Index Error = (on arc = subtract, off arc = add)
Observed Alt =
Dip = (always subtract)
Apparent Alt =
Refraction = (always subtract)
Add. Corr’n = (add or subtract – see almanac)
True altitude =
4. Find True Zenith Distance (TZD)
TZD = 90 – True Altitude
5. Find Calculated Zenith Distance (CZD);
Cos CZD = Cos LHA Cos Lat Cos Dec ± Sin LHA Sin Dec
6. Find intercept;
TZD ~ CZD (for away or towards, remember Tiny True Towards)
7. Find Azimuth using A, B and C (A is named opp to latitude unless LHA between
90˚ and 270˚
For east or west, East = LHA between 180˚ and 360˚, West = LHA 000˚ and 180˚
8. Position Line runs at 90˚ to azimuth.
9. Draw scale picture to find D’lat and departure, then find D’long.
10. Apply D’Lat and D’Long to initial position to get Intercept Position.
Rendezvous (Sunrise);
1. Find LMT of sunrise for initial position of vessel that will maintain course and speed (by interpolation).
2. Find GMT of sunrise by applying Longitude in time (+ West, - East )
3. Find time interval
Initial GMT =
GMT Sunrise = (+)
Time interval =
4. Apply time interval to vessel’s course and speed to get D’Lat and D’Long.
= 1st DR Position.
5. Using 1st DR position, find new LMT for sunrise.
6. Find new GMT for sunrise.
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dariusz.lipinski