General Navigation Formulae

General Navigation Formulae Plain Sailing: Departure (Dep) = Departure = Difference in Latitude (D’Lat) Difference in Latitude (D’Lat) =

Difference in Longitude Tangent Course

(D’Long)

X Cosine {Mean} Latitude

Distance X Cosine Course

Mercator Sailing: Difference in Longitude Longitude (D’Long) Difference in Meridian Parts Difference in Latitude (D’Lat)

=

Tangent Course

=

Distance X Cosine Course

Great Circle Sailing:

Cosine Distance = {Cosine Latitude A x Cosine Latitude B x Cosine Cosine D’Long} +/- {Sine Latitude A x Sine Latitude B} Add (+) If Both signs are in the same hemisphere. Subtract (-) If Both signs are in opposite hemispheres.

Initial Course:

A =

Tangent Latitude A Tangent {Difference in Longitude}

Tangent Latitude B Sine {Difference in Longitude} + C = A ( /-) B

A = Opposite Sign to Latitude of Initial Position unless 90o
B =

Tangent Initial Course =

B = Same Sign as Latitude Latitude of Final Position Position Add if the Signs are the same, subtract the lesser value from the greater value if not. The sign used used is the same as the greater of the two values.

1 C x Cosine Latitude A

(N) or (S) Depending on the the greater sign of A or B above above Sign {E} if 180 o
Final Course:

A =

Tangent Latitude B Tangent {Difference in Longitude}

B =

A = Opposite Sign to Latitude of Initial Position, unless 90o
Tangent Latitude A B = Same Sign as Latitude Latitude of Final Position Position Sine {Difference in Longitude} C = A ( +/-) B Add if the Signs are the same, subtract the lesser value from the greater value if  not. The sign used is the same as the greater of the two values.

Tangent Initial Course =

1 C x Cosine Latitude A

(N) or (S) Depending on the the greater sign of A or B above above Sign {E} if 180 o
Napiers Rules:

PV

AV

P

{Used to Find The Vertex, Distance Between Positions and the Vertex and D’long}. Sine Mid Part = Cosine Opposite x Cosine Opposite or  Sine Mid Part = Tangent Adjacent x Tangent Adjacent Adjacent

A

The Compliment is simply 90 o -

PA Be aware that if the difference in Longitude is less than 90 o and the difference in Latitude is more than 10 o , then the vertex could lie outside the Great Circle course line between positions. Also the Vertices may lie outside the Great Circle course line if the two positions are in opposite hemispheres. Also when positions are in opposite hemispheres the compliment of the course at the equator is the latitude of the each vertex. Finally the Longitude of each Vertex must be 90o away from the  position where the ship crosses the equator.

Celestial Navigation: Almanac Calculation Sextant Altitude Index Error (+/-) Observed Altitude Height of Eye (dip) Apparent Altitude Total Correction( +/-) True Altitude

Norries Calculation Sextant Altitude Index Error (+/-) Observed Altitude Height of Eye (dip) Apparent Altitude Refraction (-Ve ) Semi-Diameter (-Ve if Upper Limb, +Ve if Lower Limb) Parallax (+Ve) True Altitude

True Zenith Distance = 90 o - True Altitude Calculated Zenith Distance = {Cosine LHA x Cosine Latitude x Cosine Cosine Declination} +/- {Sine Latitude x Sine Declination} Declination} Add (+) If Both signs for Latitude and Declination are in the same hemisphere. Subtract (-) If Both signs for Latitude and Declination are in opposite hemispheres.

For Zenith Distance use {True Tiny Towards} if TZDCZD. If Comparing Altitudes Remember Calculated Altitude = 90 o - Calculated Zenith Distance. Azimuth:

A =

Tangent Latitude A Tangent {Difference in Longitude}

A = Opposite Sign to Latitude of Initial Position unless 90o
B =

Tangent Latitude B B = Same Sign as Latitude Latitude of Final Position Position Sine {Difference in Longitude} C = A ( +/-) B Add if the Signs are the same, subtract the lesser value from the greater value if  not. The sign used is the same as the greater of the two values.

Azimuth =

Tangent -1

1 C x Cosine Latitude

(N) or (S) Depending on the the greater sign of A or B above above Sign {E} if 180 o
Amplitudes:

Sine Amplitude = Sine Declination Cosine Latitude

Useful Nautical Almanac Formulae:

Universal Time (UTC) = Local Mean Time (LMT) + Longitude West (Expressed in Time) or  (UTC = GMT) Local Mean Time (LMT) - Longitude East (Expressed in Time) Longitude in Time = Longitude 15 Local Hour Angle

of a Body Body = Greenwich Greenwich Hour Angle (GHA) of Body + Longitude East or  Greenwich Hour Angle (GHA) of Body - Longitude West West Local Hour Angle (LHA) of a Star = Greenwich Greenwich Hour Angle Angle (GHA) of Aries ( ٢ ) + Sidereal Hour Angle Angle or

(LHA)

Greenwich Hour Angle

(GHA)

(SHA)

+ Longitude East

of Aries ( ٢ ) + Sidereal Hour Angle (SHA) - Longitude West

Meridian Passage and Rendezvous Problems Using the Second Approximation Method: When Calculating the time of Rendezvous at Sun Rise, Sunset, {Civil or Nautical} Twilights with another vessel and Meridian Passage; convert local time into Universal Time for your position. Then interpolate from the Nautical Almanac daily pages the time for Sun Rise, Sunset, {Civil or Nautical} Twilights or Meridian Passage.  Now calculate the time difference between the two times – This will give you a steaming time to calculate an approximate steaming distance and calculate a 1 st [approximate] rendezvous position positi on for the given speed and course of your vessel, using Parallel Sailing Formulae. Reinterpolate the Time of Sun Rise, Sunset, {Civil or Nautical} Twilights and Meridian Passage from the Daily Pages for this 1 st Approximate  position. Re-iterate Re-iterat e the Steaming time and Steaming distance then re-calculate a 2nd Approximate position using Parallel Sailing Formulae. Finally for this 2 nd Approximate Position, determine the Time of Sun Rise, Sunset, {Civil or Nautical} Twilights and Meridian Passage from the Daily Pages of the Almanac.