D. Latitudes and Departures
1. Definition; Equations
Latitude is the northsouth component of a line; departure the eastwest. North latitudes are positive, South are negative; similarly East departures are positive, West are negative.
(a) 
(b) 
Figure D1 Latitudes and Departures 
Latitude (Lat) and Departure (Dep) are computed from:
Equations D1 and D2 
Dir can be either a bearing angle, Figure D2(a), or azimuth angle, Figure D2(b).
(a) 
(b) 
Figure D2 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 D3 Quadrants 
An azimuth angle ranges from 0° to 360°, so the sine and cosine return the correct signs on the Lat and Dep.
Examples
Figure D4 NE Azimuth 

Figure D5 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 
(a)  (b) 
Figure D6 Reverse Latitude and Departure 
2. Closure
On a closed loop traverse, the sum of the Lats should equal zero as should the sum of the Deps.
Closure condition:


Figure D7 Perfect Closed Loop 
Most traverses won’t close perfectly due to the presence of errors. If the field crew followed careful procedures and compensated distance measurements for atmospheric conditions, only random errors prevent achieving perfect closure:event achieving perfect closure:
Actual closure:


Figure D8 Cumulative Random Errors 
Figure D9 Linear Closure 
The Linear Closure, LC, is the total misclosure distance. It is computed from the Lat and Dep errors:
Equation D7 
Traverse precision is a function of the linear closure and the total survey distance. The precision is stated as a relative value, usually as 1 unit of error per D units measured:
Equation D8 
What is allowable minimum precision? It depends on the survey purpose. Traditional first order surveys have a required precision of 1/100,000; in Wisconsin a property survey must be at least 1/3000. In order to achieve a formal minimum precision, specific equipment and field procedures must be used which match that precision requirement. Achieving a minimum precision should be by design not by accident.
3. Examples
In the following examples shown, all calculations are shown with an additional significant figure. Because these are generally intermediate computations, carrying an additional digit minimizes roundoff error in subsequent calculations.
When reporting results of an intermediate calculation, those should be stated to the correct number of significant figures so as not to imply an accuracy beyond that of the measurements.
a. Traverse with bearings
Lat and Dep will always compute as positive; must assign correct mathematical sign based on the bearing quadrant.


Figure D10 Bearing Traverse 
Line AB
Because the bearing is South and West, the Lat and Dep are 176.357' and 438.548' respectively.
Line BC
Because the bearing is North and West, the Lat and Dep are +203.395' and 73.093' respectively.
Line CD
Because the bearing is North and East, the Lat and Dep are +192.357' and +198.651' respectively.
Line DA
Because the bearing is South and East , the Lat and Dep are 219.312' and +313.065' respectively
In tabular form:
Line 
Bearing 
Length (ft) 
Lat (ft) 
Dep (ft) 
AB 
S 68°05'35"W 
472.68 
176.357 
438.548 
BC 
N 19°46'00"W 
216.13 
+203.395 
73.093 
CD 
N 45°55'20"E 
276.52 
+192.357 
+198.651 
DA 
S 54°59'15"E 
382.24 
219.312 
+313.065 
sums: 
1347.57 
+0.083 
+0.075 

Distance 
Lat err 
Dep err 


too far N 
too far E 
b. Traverse with azimuths
Lat and Dep will always compute directly with the correct sign when using azimuths. 

Figure D11 Azimuth Traverse 
Line ST
Line TU
Line UV
Line VS
Line 
Azimuth 
Length (ft) 
Lat (ft) 
Dep (ft) 
ST 
309°05'38" 
347.00 
+218.816 
269.311 
TU 
258°34'22" 
364.55 
72.226 
357.324 
UV 
128°04'44" 
472.74 
291.560 
+372.123 
VS 
60°21'26" 
292.94 
+144.885 
+254.602 
sums: 
1477.23 
0.085 
+0.090 

Distance 
Lat err 
Dep err 

too far S 
too far E 
c. Crossing Traverse
A foursided parcel has two obstructed lines.
Figure D12 Parcel Boundaries 
In order to create a closed traverse, the survey crew measures a crossing traverse which connects all four points.
As long as a traverse closes back on its beginning point, the closer condition is still: regardless of how many times it may cross itself.
Given this traverse data, determine its closure and precision.


Figure D13 
Rather than write out each Lat and Dep computation separately, we can simply set up the table and record the computations in it.
Line 
Azimuth 
Length (ft) 
Lat (ft) 
Dep (ft) 
EF 
133°02'45" 
455.03 
310.780 
+332.737 
FG 
24°33'35" 
228.35 
+207.691 
+94.912 
GH 
241°05'15" 
422.78 
204.403 
370.084 
HE 
349°25'20" 
213.85 
+307.534 
57.430 
sums: 
1419.28 
+0.042 
+0.135 

Distance 
Lat err 
Dep err 

too far N 
Too far E 