D. Latitudes and Departures

1. Definition; Equations

Latitude is the north-south component of a line; departure the east-west. North latitudes are positive, South are negative; similarly East departures are positive, West are negative.


The latitude of line AB is North (+),
its departure is East (+).

The latitude of line CD is South (-),
its departure is West (-).

Figure D-1
Latitudes and Departures

Latitude (Lat) and Departure (Dep) are computed from:

img4        Equations D-1 and D-2


Dir can be either a bearing angle, Figure D-2(a), or azimuth angle, Figure D-2(b). 

img15 img16

LatAB is North
DepAB is East

LatBA is South
DepBA is West

Figure D-2
Bearings or Azimuths


Because a bearing angle never exceeds 90°, the Lat and Dep equations will always return positive values.


Sin(0°) to Sin(90°) ranges from 0 to +1.0

Cos(0°) to Cos(90°) ranges from +1.0 to 0

The correct mathematical sign for the Lat and Dep come from the bearing quadrant.

A bearing of S 47°35' E has a negative Lat (South) and a positive Dep (East).

Figure D-3


An azimuth angle ranges from 0° to 360°, so the sine and cosine return the correct signs on the Lat and Dep.



d 05

  Figure D-4
NE Azimuth



  Figure D-5
SW Azimuth


Reversing a line direction results in the same magnitude Lat and Dep but reversed signs:

Line A to B Line B to A
img22 img21
 (a) (b) 
Figure D-6
Reverse Latitude and Departure