LATITUDE AND DEPARTURE
Latitude of any line – is the projection on a north and south lines. It
may be called as north or positive latitude and south or negative
latitude.
Departure of any line – is the projection on the east and west line.
West departure is sometimes called negative departure and East
departure is sometimes called positive departure.
Line AB has its latitude AC and departure BC.
𝐶 𝐷𝐸𝑃𝐴𝑅𝑇𝑈𝑅𝐸 The angle 𝜃 is the bearing of the line AB.
𝐵
𝐵𝐶 = 𝐴𝐵 𝑠𝑖𝑛𝜃
𝐿𝐴𝑇𝐼𝑇𝑈𝐷𝐸
Departure = Distance x Sin 𝜃
𝐴𝐶 = 𝐴𝐵 𝑐𝑜𝑠𝜃
𝜃
Latitude = Distance x Cos bearing
𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒
𝐴 𝐷𝑖𝑠𝑡 = 𝐷𝑖𝑠𝑡 =
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
G.W.B.Paule
ERROR OF CLOSURE
In any closed traverse, there is always an error. No survey is
geometrically perfect until proper adjustments are made. For a
closed traversed, the sum of the north and south latitudes should
always be zero.
BALANCING A SURVEY
1. Compass Rule – the correction to be applied to the latitude or
departure of any course is the total correction in latitude or
departure as the length of the course is to the length of the
traverse.
2. Transit Rule – the correction to be applied to the latitude or
departure of any course is the total correction in latitude or
departure as the latitude or departure of that course is to the
arithmetical sum of all the latitudes or departures in the traverse
without regards to sign.
𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑐𝑙𝑜𝑠𝑢𝑟𝑒 = 𝐿2 + 𝐷 2
𝐸𝑟𝑟𝑜𝑟 𝑜𝑓 𝐶𝑙𝑜𝑠𝑢𝑟𝑒
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑐𝑜𝑢𝑟𝑠𝑒𝑠
𝐿 = 𝑒𝑟𝑟𝑜𝑟 𝑖𝑛 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒
𝐷 = 𝑒𝑟𝑟𝑜𝑟 𝑖𝑛 𝑑𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒
COMPUTATION OF AREA
1. Area by Triangle Method
2
𝛼 𝐴 = 𝐴1 + 𝐴2 + 𝐴3
𝑑1
1
𝑑2 𝐴1 = 𝑑1 𝑑2 𝑠𝑖𝑛𝛼
1 2
1 1
𝑑3 𝐴2 = 𝑑 𝑑 𝑠𝑖𝑛𝛽
𝛽 2 3 4
1
2 3 𝐴3 = 𝑑5 𝑑6 𝑠𝑖𝑛𝜃
2
𝑑4
𝑑5 3
𝜃
5 𝑑6 4
G.W.B.Paule
2. Area by Double Meridian Distance
- DMD of the first course is equal to the departure of that course.
- DMD of any other course is equal to the DMD of the preceding course, plus the
departure of the preceding course plus the departure of the course itself.
- DMD of the last course is numerically equal to the departure of the last course but
opposite in sign.
Computing Area by DMD Method:
1. Compute the latitude and departures of all courses.
2. Compute the error of closure in latitude and departures.
3. Balance the latitudes and departures by applying either the transit rule or the
compass rule.
4. Compute for the DMD of all courses.
5. Compute the double areas by multiplying each DMD by the corresponding latitude.
6. Determine the algebraic sum of the double areas.
7. Divide the algebraic sum of the double area to obtain the area of the whole tract.
𝑫𝒐𝒖𝒃𝒍𝒆 𝑨𝒓𝒆𝒂 = 𝑫𝑴𝑫 𝒙 𝑳𝒂𝒕𝒊𝒕𝒖𝒅𝒆 G.W.B.Paule
Example: Area by Double Meridian Distance
𝑫𝒐𝒖𝒃𝒍𝒆 𝑨𝒓𝒆𝒂 = 𝑫𝑴𝑫 𝒙 𝑳𝒂𝒕𝒊𝒕𝒖𝒅𝒆
Lines LAT DEP DMD DOUBLE AREA
1-2 +60 -30 -30 -30 (60) = -1800
2-3 -20 +20 (-30) + (-30) + (+20) = -40 -40 (-20) = +800
3-4 -80 +60 (-40) + (+20) + (+60) = +40 +40 (-80) = -3200
4-1 +40 -50 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑔𝑛 +50 +50 (40) = +2000
2A = -2200
A = -1100 m^2
- DMD of the first course is equal to the departure of that course.
- DMD of any other course is equal to the DMD of the preceding course, plus the departure of the preceding
course plus the departure of the course itself.
- DMD of the last course is numerically equal to the departure of the last course but opposite in sign.
3. Area by Double Parallel Distance
1. DPD of the first course is equal to the latitude of that course.
2. DPD of any other course is equal to the DPD of the preceding course,
plus the latitude of the preceding course, plus the latitude of the
course itself.
3. DPD of the last course is numerically equal to the latitude of the last
course but opposite in sign.
𝑫𝒐𝒖𝒃𝒍𝒆 𝑨𝒓𝒆𝒂 = 𝑫𝒐𝒖𝒃𝒍𝒆 𝑷𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒙 𝑫𝒆𝒑𝒂𝒓𝒕𝒖𝒓𝒆
Example: Area by Double Parallel Distance
Lines LAT DEP DPD DOUBLE AREA
1-2 +60 -30 +60 60 (-30) = -1800
2-3 -20 +20 (+60) + (+60) + (-20) = +100 100 (20) = +2000
3-4 -80 +60 (-20) + (+100)+ (-80) = 0 0 (60) = 0
4-1 +40 -50 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑔𝑛 -40 -40 (50) = +2000
2A = -2200
A = 1100 m^2
- DPD of the first course is equal to the latitude of that course.
- DPD of any other course is equal to the DPD of the preceding course, plus the latitude of the
preceding course, plus the latitude of the course itself.
- DPD of the last course is numerically equal to the latitude of the last course but opposite in sign.
Sample Problem 1:
From the field notes of a closed traverse shown below, adjust the
traverse.
LINES BEARINGS DISTANCE
AB Due North 400 m
BC 𝑁 45° 𝐸 800 m
CD 𝑆 60° 𝐸 700 m
DE 𝑆 20° 𝑊 600 m
EA 𝑆 86°59′ 𝑊 966.34 m
a. Compute the correction of latitude on line CD using transit rule.
b. Compute the linear error of closure.
c. Compute the relative error of precision.
