CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
1/13
Surveying - Traverse
Introduction Almost all surveying requires some calculations to reduce measurements into a more useful form for determining distance, earthwork volumes, land areas, etc. A traverse is developed by measuring the distance and angles between points that found the boundary of a site We will learn several different techniques to compute the area inside a traverse
Surveying - Traverse
Distance - Traverse Methods of Computing Area A simple method that is useful for rough area estimates is a graphical method In this method, the traverse is plotted to scale on graph paper, and the number of squares inside the traverse are counted
B
A C D
Distance - Traverse
Distance - Traverse
Methods of Computing Area B a A
Area ABC
b a
Methods of Computing Area B
1 ac sin 2
a A
c
C
b
a
Area ABD
1 ad sin 2
Area BCD
1 bc sin 2
b C
d c D
Area ABCD Area ABD Area BCD
CIVL 1112
Surveying - Traverse Calculations
Distance - Traverse
Surveying - Traverse
Methods of Computing Area B
b
A
Area ABE
c
a
Balancing Angles
C
a
b D
e
2/13
Area CDE
d
1 ae sin 2
Before the areas of a piece of land can be computed, it is necessary to have a closed traverse The interior angles of a closed traverse should total:
1 cd sin 2
(n - 2)(180°) where n is the number of sides of the traverse
E
To compute Area BCD more data is required
Surveying - Traverse
Surveying - Traverse
Balancing Angles
Balancing Angles A
Error of closure
B D
A surveying heuristic is that the total angle should not vary from the correct value by more than the square root of the number of angles measured times the precision of the instrument For example an eight-sided traverse using a 1’ transit, the maximum error is:
1' 8 2.83' 3' C
Angle containing mistake
Surveying - Traverse
Surveying - Traverse
Balancing Angles
Latitudes and Departures
If the angles do not close by a reasonable amount, mistakes in measuring have been made
The closure of a traverse is checked by computing the latitudes and departures of each of it sides
If an error of 1’ is made, the surveyor may correct one angle by 1’ If an error of 2’ is made, the surveyor may correct two angles by 1’ each If an error of 3’ is made in a 12 sided traverse, the surveyor may correct each angle by 3’/12 or 15”
N
N
B
Latitude AB
Bearing E
W Bearing
A
W C
Departure AB Latitude CD
S
Departure CD
S
D
E
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
3/13
Surveying - Traverse
Latitudes and Departures
Error of Closure
The latitude of a line is its projection on the north–south meridian
Consider the following statement:
N Latitude AB
E
W Bearing
A
“If start at one corner of a closed traverse and walk its lines until you return to your starting point, you will have walked as far north as you walked south and as far east as you have walked west”
The departure of a line is its projection on the east– west line
B
Departure AB
A northeasterly bearing has: + latitude and + departure
latitudes = 0
Therefore --
and
departures = 0
S
Surveying - Traverse
Surveying - Traverse
Error of Closure
Error of Closure
When latitudes are added together, the resulting error is called the error in latitudes (EL)
If the measured bearings and distances are plotted on a sheet of paper, the figure will not close because of EL and ED
The error resulting from adding departures together is called the error in departures (ED)
B
Error of closure
ED
Eclosure
EL A
C
Latitudes and Departures - Example
Precision
2
ED
2
Eclosure perimeter
Typical precision: 1/5,000 for rural land, 1/7,500 for suburban land, and 1/10,000 for urban land
D
Surveying - Traverse
EL
Surveying - Traverse Latitudes and Departures - Example
A N N 42° 59’ E
S 6° 15’ W
234.58’
Departure AB
189.53’
B
W (189.53 ft.)sin(615 ') 20.63 ft. A
W
E
E S 29° 38’ E 142.39’ 175.18’
N 12° 24’ W
S 6° 15’ W
Latitude AB
189.53 ft.
S (189.53 ft.)cos(615 ') 188.40 ft.
175.18’
D N 81° 18’ W
C
B
S
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
4/13
Surveying - Traverse
Latitudes and Departures - Example
Latitudes and Departures - Example Bearing
Side
N Departure BC
E (175.18 ft.)sin(2938 ') 86.62 ft. B
W
AB BC CD DE EA
degree
m inutes
6 29 81 12 42
15 38 18 24 59
S S N N N
Length (ft.)
Latitude
Departure
189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
W E W W E
E 175.18 ft.
Latitude BC
S 29° 38’ E
S (175.18 ft.)cos(2938') 152.27 ft.
C S
Surveying - Traverse
Surveying - Traverse
Latitudes and Departures - Example Bearing
Side AB BC CD DE EA
Eclosure
S S N N N
EL
2
degree
m inutes
6 29 81 12 42
15 38 18 24 59
ED
Length (ft.)
Latitude
Departure
189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
A S 77° 10’ E
W E W W E
0.079
2
Group Example Problem 1
2
0.163 0.182 ft.
N 29° 16’ E
651.2 ft.
B 660.5 ft.
S 38° 43’ W
2
0.182 ft. Eclosure Precision 939.46 ft. perimeter
1 5,176
D
491.0 ft.
826.7 ft.
N 64° 09’ W C
Surveying - Traverse
Surveying - Traverse Balancing Latitudes and Departures
Group Example Problem 1
Balancing the latitudes and departures of a traverse attempts to obtain more probable values for the locations of the corners of the traverse Bearing
Side AB BC CD DE
S S N N
degree
m inutes
77 38 64 29
10 43 9 16
Length (ft.) E W W E
651.2 826.7 491.0 660.5
Latitude
Departure
A popular method for balancing errors is called the compass or the Bowditch rule The “Bowditch rule” as devised by Nathaniel Bowditch, surveyor, navigator and mathematician, as a proposed solution to the problem of compass traverse adjustment, which was posed in the American journal The Analyst in 1807.
CIVL 1112
Surveying - Traverse Calculations
5/13
Surveying - Traverse
Surveying - Traverse
Balancing Latitudes and Departures
Balancing Latitudes and Departures A
The compass method assumes: N 42° 59’ E
1) angles and distances have same error 2) errors are accidental
S 6° 15’ W
234.58 ft.
189.53 ft.
B
The rule states: E
S 29° 38’ E
“The error in latitude (departure) of a line is to the total error in latitude (departure) as the length of the line is the perimeter of the traverse”
142.39 ft. 175.18 ft.
N 12° 24’ W
175.18 ft.
D N 81° 18’ W
Surveying - Traverse
C
Surveying - Traverse
Latitudes and Departures - Example
Latitudes and Departures - Example
Recall the results of our example problem
Recall the results of our example problem
Side
Length (ft)
Bearing
AB BC CD DE EA
S S N N N
degree
m inutes
6 29 81 12 42
15 38 18 24 59
W E W W E
Latitude
Departure
Side
189.53 175.18 197.78 142.39 234.58
Bearing
AB BC CD DE EA
S S N N N
degree
m inutes
6 29 81 12 42
15 38 18 24 59
W E W W E
Length (ft)
Latitude
Departure
189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
Surveying - Traverse
Surveying - Traverse
Balancing Latitudes and Departures
Balancing Latitudes and Departures
N
N Latitude AB
Departure AB
S (189.53 ft.)cos(6 15 ') 188.40 ft.
A
W
E
Correction in Lat AB LAB EL perimeter
S 6° 15’ W 189.53 ft.
B
W (189.53 ft.)sin(615 ') 20.63 ft. A
W
Correction in Lat AB
EL LAB
189.53 ft.
B
Correction in Lat AB
939.46 ft.
Correction in Dep AB LAB ED perimeter
S 6° 15’ W
Correction in Dep AB
perimeter
S
0.079 ft. 189.53 ft.
E
0.016 ft.
ED LAB perimeter
S
Correction in Dep AB
0.163 ft. 189.53 ft. 939.46 ft.
0.033 ft.
CIVL 1112
Surveying - Traverse Calculations
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Surveying - Traverse
Surveying - Traverse
Balancing Latitudes and Departures
Balancing Latitudes and Departures
N
N Latitude BC
Departure BC
S (175.18 ft.)cos(29 38 ') 152.27 ft.
B
W
E
Correction in LatBC LBC EL perimeter
175.18 ft.
S 29° 38’ E
Correction in LatBC C
S
Correction in LatBC
EL LBC
E (175.18 ft.)sin(2938 ') 86.62 ft. B
W
E
S 29° 38’ E
939.46 ft.
ED LBC
Correction in DepBC
perimeter
0.079 ft. 175.18 ft.
Correction in DepBC LBC ED perimeter
175.18 ft.
perimeter
C S
0.015 ft.
Correction in DepBC
0.163 ft. 175.18 ft.
0.030 ft.
939.46 ft.
Surveying - Traverse
Surveying - Traverse
Balancing Latitudes and Departures
Balancing Latitudes and Departures
Length (ft.) Latitude 189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
Departure -20.634 86.617 -195.504 -30.576 159.933 -0.163
Corrections Latitude Departure 0.016 0.015
Balanced Latitude Departure
0.033 0.030
Length (ft.) Latitude 189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
Departure -20.634 86.617 -195.504 -30.576 159.933 -0.163
Corrections Latitude Departure 0.016 0.015
0.033 0.030
Balanced Latitude Departure -188.388 -152.253
Corrected latitudes and departures
Corrections computed on previous slides
Surveying - Traverse
Surveying - Traverse
Balancing Latitudes and Departures
Balancing Latitudes and Departures
Length (ft.) Latitude 189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
Departure -20.634 86.617 -195.504 -30.576 159.933 -0.163
Corrections Latitude Departure 0.016 0.015 0.017 0.012 0.020
-20.601 86.648
0.033 0.030 0.034 0.025 0.041
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627 0.000
No error in corrected latitudes and departures
-20.601 86.648 -195.470 -30.551 159.974 0.000
Combining the latitude and departure calculations with corrections gives: Side
Corrections Length (ft.) Latitude Departure Latitude Departure
Bearing
Balanced Latitude Departure
degree m inutes
AB BC CD DE EA
S S N N N
6 29 81 12 42
15 38 18 24 59
W E W W E
189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
0.016 0.015 0.017 0.012 0.020
0.033 0.030 0.034 0.025 0.041
-188.388 -152.253 29.933 139.080 171.627 0.000
-20.601 86.648 -195.470 -30.551 159.974 0.000
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
7/13
Surveying - Traverse
Group Example Problem 2
Group Example Problem 3
Balance the latitudes and departures for the following traverse.
In the survey of your assign site in Project #3, you will have to balance data collected in the following form:
Corrections Balanced Length (ft) Latitude Departure Latitude Departure Latitude Departure
N 600.0 450.0 750.0 1800.0
450.00 -285.00 -164.46 0.54
339.00 259.50 -599.22 -0.72
B
N 69° 53’ E
A
51° 23’
713.93 ft. 105° 39’ 606.06 ft.
781.18 ft.
78° 11’
C
124° 47’ 391.27 ft.
D
Surveying - Traverse
Surveying - Traverse Calculating Traverse Area
Group Example Problem 3 In the survey of your assign site in Project #3, you will have to balance data collected in the following form: Side
Corrections Length (ft.) Latitude Departure Latitude Departure
Bearing
Balanced Latitude Departure
degree m inutes
AB N BC CD DA
69
53
E
713.93 606.06 391.27 781.18
Eclosure =
Precision =
The meridian distance of a line is the east–west distance from the midpoint of the line to the reference meridian The meridian distance is positive (+) to the east and negative (-) to the west
ft. 1
Surveying - Traverse Calculating Traverse Area N
Surveying - Traverse Calculating Traverse Area
A N 42° 59’ E
S 6° 15’ W
234.58 ft.
189.53 ft.
B E S 29° 38’ E 142.39 ft.
Reference Meridian
The best–known procedure for calculating land areas is the double meridian distance (DMD) method
175.18 ft.
N 12° 24’ W 175.18 ft.
D N 81° 18’ W
C
The most westerly and easterly points of a traverse may be found using the departures of the traverse Begin by establishing a arbitrary reference line and using the departure values of each point in the traverse to determine the far westerly point
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
Surveying - Traverse
Calculating Traverse Area Length (ft.) Latitude 189.53 175.18 197.78 142.39 234.58 939.46
Departure
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
-20.601 -195.470
D -30.551
E
D
Calculating Traverse Area
Corrections Latitude Departure 0.016 0.015 0.017 0.012 0.020
B
0.033 0.030 0.034 0.025 0.041
N
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627 0.000
-20.601 86.648 -195.470 -30.551 159.974 0.000
Reference Meridian
86.648
189.53 ft.
142.39 ft.
N 12° 24’ W
S 29° 38’ E 175.18 ft.
D N 81° 18’ W
Surveying - Traverse
C
Surveying - Traverse
DMD Calculations
DMD Calculations The meridian distance of line EA is:
A
N
Meridian distance of line AB
N
B
The meridian distance of line AB is equal to:
A
the meridian distance of EA + ½ the departure of line EA + ½ departure of AB
B
The DMD of line AB is twice the meridian distance of line AB
A
E
E C
S 6° 15’ W
234.58 ft.
E C C
Point E is the farthest to the west
A
D
N 42° 59’ E
175.18 ft.
E
Reference Meridian
A
B
A
B
159.974
N
8/13
E
DMD of line EA is the departure of line
Surveying - Traverse DMD Calculations
N
Meridian distance of line AB
A
B E
Surveying - Traverse DMD Calculations
The DMD of any side is equal to the DMD of the last side plus the departure of the last side plus the departure of the present side
Side AB BC CD DE EA
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
DMD -20.601
The DMD of line AB is departure of line AB
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
Surveying - Traverse
DMD Calculations Side AB BC CD DE EA
DMD Calculations
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
-20.601 + 86.648 + -195.470 -30.551 159.974
Side DMD -20.601 45.447
The DMD of line BC is DMD of line AB + departure of line AB + the departure of line BC
Surveying - Traverse
AB BC CD DE EA
DMD -20.601 -20.601 45.447 86.648 -195.470 + -63.375 -30.551 + -289.397 159.974
Surveying - Traverse
AB BC CD DE EA
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
DMD -20.601 -20.601 45.447 86.648 -63.375 -195.470 -30.551 + -289.397 159.974 + -159.974
The DMD of line EA is DMD of line DE + departure of line DE + the departure of line EA
Traverse Area - Double Area The sum of the products of each points DMD and latitude equal twice the area, or the double area
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
DMD -20.601 45.447 -63.375
Surveying - Traverse
DMD Calculations
AB BC CD DE EA
-20.601 86.648 + -195.470 + -30.551 159.974
The DMD of line CD is DMD of line BC + departure of line BC + the departure of line CD
Side
The DMD of line DE is DMD of line CD + departure of line CD + the departure of line DE
Side
-188.388 -152.253 29.933 139.080 171.627
DMD Calculations
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
AB BC CD DE EA
Balanced Latitude Departure
Surveying - Traverse
DMD Calculations Side
9/13
-20.601 86.648 -195.470 -30.551 159.974
DMD -20.601 45.447 -63.375 -289.397 -159.974
Notice that the DMD values can be positive or negative
Side AB BC CD DE EA
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
DMD Double Areas -20.601 3,881 45.447 -63.375 -289.397 -159.974
The double area for line AB equals DMD of line AB times the latitude of line AB
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
Surveying - Traverse
Traverse Area - Double Area
Traverse Area - Double Area
The sum of the products of each points DMD and latitude equal twice the area, or the double area Side AB BC CD DE EA
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
DMD Double Areas -20.601 3,881 -6,919 45.447 -63.375 -289.397 -159.974
Surveying - Traverse
Balanced Latitude Departure -20.601 86.648 -195.470 -30.551 159.974
Surveying - Traverse
Balanced Latitude Departure
1 acre = 43,560 ft.2
-20.601 86.648 -195.470 -30.551 159.974
The sum of the products of each points DMD and latitude equal twice the area, or the double area
AB BC CD DE EA
-188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
DMD Double Areas -20.601 3,881 -6,919 45.447 -1,897 -63.375 -40,249 -289.397 -27,456 -159.974
The double area for line EA equals DMD of line EA times the latitude of line EA
The sum of the products of each points DMD and latitude equal twice the area, or the double area Side
DMD Double Areas -20.601 3,881 45.447 -6,919 -63.375 -1,897 -289.397 -40,249 -159.974 -27,456 2 Area = -72,641
Area =
Balanced Latitude Departure
Traverse Area - Double Area
The sum of the products of each points DMD and latitude equal twice the area, or the double area
-188.388 -152.253 29.933 139.080 171.627
DMD Double Areas -20.601 3,881 -6,919 45.447 -1,897 -63.375 -289.397 -159.974
Surveying - Traverse
Traverse Area - Double Area
AB BC CD DE EA
-20.601 86.648 -195.470 -30.551 159.974
The double area for line CD equals DMD of line CD times the latitude of line CD
Side DMD Double Areas -20.601 3,881 -6,919 45.447 -1,897 -63.375 -40,249 -289.397 -159.974
The double area for line DE equals DMD of line DE times the latitude of line DE
Side
-188.388 -152.253 29.933 139.080 171.627
Traverse Area - Double Area
The sum of the products of each points DMD and latitude equal twice the area, or the double area
-188.388 -152.253 29.933 139.080 171.627
AB BC CD DE EA
Balanced Latitude Departure
Surveying - Traverse
Traverse Area - Double Area
AB BC CD DE EA
The sum of the products of each points DMD and latitude equal twice the area, or the double area Side
The double area for line BC equals DMD of line BC times the latitude of line BC
Side
10/13
36,320 ft.2 0.834 acre
AB BC CD DE EA
Balanced Latitude Departure -188.388 -152.253 29.933 139.080 171.627
1 acre = 43,560 ft.2
-20.601 86.648 -195.470 -30.551 159.974
DMD Double Areas -20.601 3,881 45.447 -6,919 -63.375 -1,897 -289.397 -40,249 -159.974 -27,456 2 Area = -72,641
Area =
36,320 ft.2 0.834 acre
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse Traverse Area - Double Area
11/13
Surveying - Traverse Traverse Area - Double Area
The word "acre" is derived from Old English æcer (originally meaning "open field", cognate to Swedish "åker", German acker, Latin ager and Greek αγρος (agros).
The word "acre" is derived from Old English æcer (originally meaning "open field", cognate to Swedish "åker", German acker, Latin ager and Greek αγρος (agros).
The acre was selected as approximately the amount of land tillable by one man behind an ox in one day.
A long narrow strip of land is more efficient to plough than a square plot, since the plough does not have to be turned so often.
This explains one definition as the area of a rectangle with sides of length one chain (66 ft.) and one furlong (ten chains or 660 ft.).
The word "furlong" itself derives from the fact that it is one furrow long.
Surveying - Traverse Traverse Area - Double Area The word "acre" is derived from Old English æcer (originally meaning "open field", cognate to Swedish "åker", German acker, Latin ager and Greek αγρος (agros).
Surveying - Traverse Traverse Area – Example 4 Find the area enclosed by the following traverse
Side
Balanced Latitude Departure DMD
AB BC CD DE EA
600.0 100.0 0.0 -400.0 -300.0
Double Areas
200.0 400.0 100.0 -300.0 -400.0 2 Area =
1 acre = 43,560 ft.2
Surveying - Traverse DPD Calculations
Area =
ft. 2 acre
Surveying - Traverse Rectangular Coordinates
The same procedure used for DMD can be used the double parallel distances (DPD) are multiplied by the balanced departures
Rectangular coordinates are the convenient method available for describing the horizontal position of survey points
The parallel distance of a line is the distance from the midpoint of the line to the reference parallel or east–west line
With the application of computers, rectangular coordinates are used frequently in engineering projects In the US, the x–axis corresponds to the east–west direction and the y–axis to the north–south direction
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
Surveying - Traverse
Rectangular Coordinates Example
Rectangular Coordinates Example
In this example, the length of AB is 300 ft. and bearing is shown in the figure below. Determine the coordinates of point B y
Latitude AB =300 ft. cos(4230’) = 221.183 ft.
B
In this example, it is assumed that the coordinates of points A and B are know and we want to calculate the latitude and departure for line AB y
A
Latitude AB = -400 ft.
x
Departure AB = x B – x A Departure AB = 220 ft.
x B = 200 + 202.667 = 402.667 ft.
B
Surveying - Traverse
Rectangular Coordinates Example
Rectangular Coordinates Example y
Consider our previous example, determine the x and y coordinates of all the points A
Side AB BC CD DE EA
B
E
D
-20.601 86.648 -195.470 -30.551 159.974
x
C
Surveying - Traverse Rectangular Coordinates Example y
A
E
x coordinates E = 0 ft.
B D
x
C
Side AB BC CD DE EA
-188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
AB BC CD DE EA
C = B + 86.648 = 226.021 ft. D = C – 195.470 = 30.551 ft. E = D – 30.551 = 0 ft.
Surveying - Traverse Rectangular Coordinates Example y
A (159.974, 340.640)
C = 0 ft. C
x
D = C + 29.933 ft. B (139.373, 152.253)
E = D + 139.080 = 169.013 ft. Side
A = E + 159.974 = 159.974 ft. B = A – 20.601 = 139.373 ft.
Balanced Latitude Depa rture
y coordinates
B D
A
E
Balanced Latitude Depa rture -188.388 -152.253 29.933 139.080 171.627
x
Coordinates of Point B (320, -100)
y B = 300 + 221.183 = 521.183 ft.
Surveying - Traverse
y
Latitude AB = y B – y A
Coordinates of Point A (100, 300)
A
Departure AB =300 ft. sin(4230’) = 202.677 ft.
N 42 30’ E
Coordinates of Point A (200, 300)
12/13
Balanced Latitude Depa rture -188.388 -152.253 29.933 139.080 171.627
-20.601 86.648 -195.470 -30.551 159.974
(0.0, 169.013)
A = E + 171.627 = 340.640 ft.
E
B = A –188.388 = 152.252 ft. (30.551, 29.933)
D
C = B –152.252 = 0 ft. C (226.020, 0.0)
x
CIVL 1112
Surveying - Traverse Calculations
Surveying - Traverse
Surveying - Traverse Area Computed by Coordinates
Group Example Problem 5 Compute the x and y coordinates from the following balanced. Side
Balanced Length (ft.) Latitude Departure Latitude Departure
Bearing
Coordinates x y
Points
degree m inutes
AB BC CD DE EA
S S N N N
6 29 81 12 42
15 38 18 24 59
W E W W E
189.53 175.18 197.78 142.39 234.58 939.46
-188.403 -152.268 29.916 139.068 171.607 -0.079
-20.634 86.617 -195.504 -30.576 159.933 -0.163
-188.388 -152.253 29.933 139.080 171.627 0.000
13/13
-20.601 86.648 -195.470 -30.551 159.974 0.000
A B C D E
100.000
100.000
The area of a traverse can be computed by taking each y coordinate multiplied by the difference in the two adjacent x coordinates (using a sign convention of + for next side and - for last side)
Surveying - Traverse
Surveying - Traverse
Area Computed by Coordinates
Area Computed by Coordinates
y
A (159.974, 340.640)
Twice the area equals: = 340.640(139.373 – 0.0)
There is a simple variation of the coordinate method for area computation y
B (139.373, 152.253) (0.0, 169.013) E
A (159.974, 340.640)
+ 152.253(226.020 – 159.974)
x1 y1
+ 0.0(30.551 – 139.373)
x2 y2
x3 y3
x4 y4
x5 y5
B (139.373, 152.253) (30.551, 29.933)
D
+ 29.933(0.0 – 226.020)
(0.0, 169.013) E
Twice the area equals:
C (226.020, 0.0) x
+ 169.013(159.974 – 30.551) = 72,640.433 ft.2 Area = 0.853 acre
Area Computed by Coordinates There is a simple variation of the coordinate method for area computation A (159.974, 340.640)
B (139.373, 152.253)
Twice the area equals: 159.974(152.253) + 139.373(0.0) + 226.020(29.933) + 30.551(169.013) + 0.0(340.640)
(0.0, 169.013) E
(30.551, 29.933)
- 340.640(139.373) – 152.253(226.020) - 0.0(30.551) – 29.933(0.0) – 169.013(159.974)
D
= x1y2 + x2y3 + x3y4 + x4y5 + x5y1
D C (226.020, 0.0) x
- x2y1 – x3y2 – x4y3 – x5y4 – x1y5
Area = 36,320.2 ft.2
Surveying - Traverse
y
(30.551, 29.933)
C (226.020, 0.0) x
= -72,640 ft.2
Area = 36,320 ft.2
End of Surveying - Traverse Any Questions?
x1 y1