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Surveying - Traverse
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
Introduction
Surveying - 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
A
D
C
B
Distance - Traverse
Methods of Computing Area
A
C
B
a b
c
sin
Area ABC ac
Distance - Traverse
Methods of Computing Area
A
C
B
a b
c
sin
Area ABD ad
D
d
sin
Area BCD bc
Area ABCD Area ABD Area BCD
Distance - Traverse
Methods of Computing Area
A
B C
a
b
c
sin
Area ABE ae
E
d
sin
Area CDE cd
D
e
To compute Area BCD more data is required
Surveying - Traverse
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:
( n - 2)(180°)
where n is the number of sides of the traverse
Balancing Angles
Surveying - Traverse
Balancing Angles
B
D
A
C
Error of closure
Angle containing mistake
Surveying - Traverse
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:
Balancing Angles
Surveying - Traverse
If the angles do not close by a reasonable amount,
mistakes in measuring have been made
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”
Balancing Angles
E
N
S
W
Surveying - Traverse
The closure of a traverse is checked by computing the
latitudes and departures of each of it sides
Latitudes and Departures
A
E
N
B
Departure (^) AB
Latitude (^) AB
S
W
C
D
Latitude (^) CD
Departure (^) CD
Bearing
Bearing
Surveying - Traverse
E
N
S
W B
C
Latitude (^) BC
Departure (^) BC
Latitudes and Departures - Example
S 29° 38’ E
E (175.18 ft.)sin(29 38 ') 86.62 ft.
S (175.18 ft.)cos(29 38 ') 152.27 ft.
175.18 ft.
Surveying - Traverse
Latitudes and Departures - Example
Side Length (ft.) Latitude Departure degree m inutes AB S 6 15 W 189.53 -188.403 -20. BC S 29 38 E 175.18 -152.268 86. CD N 81 18 W 197.78 29.916 -195. DE N 12 24 W 142.39 139.068 -30. EA N 42 59 E 234.58 171.607 159. 939.46 -0.079 -0.
Bearing
Surveying - Traverse
Latitudes and Departures - Example
2 2
E closure EL ED
2 2 0.079 0.163 0.182 ft.
Precision perimeter
E^ closure 0.182 ft. 939.46 ft.
1
5,
Side Length (ft.) Latitude Departure degree m inutes AB S 6 15 W 189.53 -188.403 -20. BC S 29 38 E 175.18 -152.268 86. CD N 81 18 W 197.78 29.916 -195. DE N 12 24 W 142.39 139.068 -30. EA N 42 59 E 234.58 171.607 159. 939.46 -0.079 -0.
Bearing
Surveying - Traverse
Group Example Problem 1
B
A
D
C
S 77° 10’ E
651.2 ft.
S 38° 43’ W
826.7 ft.
N 64° 09’ W
491.0 ft.
N 29° 16’ E
660.5 ft.
Side Length (ft.) Latitude Departure degree minutes AB S 77 10 E 651. BC S 38 43 W 826. CD N 64 9 W 491. DE N 29 16 E 660.
Bearing
Surveying - Traverse
Group Example Problem 1
Surveying - Traverse
Balancing the latitudes and departures of a traverse
attempts to obtain more probable values for the locations
of the corners of the traverse
A popular method for balancing errors is called the
compass or the Bowditch rule
Balancing Latitudes and Departures
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.
Surveying - Traverse
The compass method assumes:
1) angles and distances have same error
2) errors are accidental
The rule states:
“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”
Balancing Latitudes and Departures
Surveying - Traverse
Balancing Latitudes and Departures
E
B
A
D
C
S 6° 15’ W
189.53’
175.18’S 29° 38’ E
N 81° 18’ W
197.78’
N 12° 24’ W
142.39’
N 42° 59’ E
234.58’
Surveying - Traverse
Latitudes and Departures - Example
Side Length (ft) Latitude Departure degree m inutes AB S 6 15 W 189. BC S 29 38 E 175. CD N 81 18 W 197. DE N 12 24 W 142. EA N 42 59 E 234.
Bearing
Recall the results of our example problem
Surveying - Traverse
Latitudes and Departures - Example
Side Length (ft) Latitude Departure degree m inutes AB S 6 15 W 189.53 -188.403 -20. BC S 29 38 E 175.18 -152.268 86. CD N 81 18 W 197.78 29.916 -195. DE N 12 24 W 142.39 139.068 -30. EA N 42 59 E 234.58 171.607 159. 939.46 -0.079 -0.
Bearing
Recall the results of our example problem
Surveying - Traverse
Balancing Latitudes and Departures
E
N
S
W A
B
S 6° 15’ W
189.53 ft.
Correction in LatAB perimeter
AB L
L
E
Correction in Lat AB perimeter
E^ L^ LAB
AB
0.079 ft. 189.53 ft. Correction in Lat 939.46 ft.
^ 0.016 ft.
Latitude (^) AB
S (189.53 ft.)cos(6 15') 188.40 ft.
Surveying - Traverse
Departure (^) AB
W (189.53 ft.)sin(6 15') 20.63 ft.
Correction in Dep AB perimeter
AB D
L
E
Correction in Dep (^) AB perimeter
E^ D^ LAB
AB
0.163 ft. 189.53 ft. Correction in Dep 939.46 ft.
^ 0.033 ft.
Balancing Latitudes and Departures
E
N
S
W A
B
S 6° 15’ W
189.53 ft.
Surveying - Traverse
Group Example Problem 2
Balance the latitudes and departures for the following
traverse.
Length (ft) Latitude Departure Latitude Departure Latitude Departure
Corrections Balanced
Surveying - Traverse
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:
B
A
D
C
51° 23’ 713.93 ft. 606.06 ft.
N 69° 53’ E
391.27 ft.
781.18 ft.
N
78° 11’ 124° 47’
105° 39’
Surveying - Traverse
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 Length (ft.) Latitude Departure Latitude Departure Latitude Departure degree m inutes AB N 69 53 E 713. BC 606. CD 391. DA 781.
E (^) closure = ft.
1
Bearing
Corrections Balanced
Precision =
Surveying - Traverse
The best-known procedure for calculating land areas is
the double meridian distance (DMD) method
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
Calculating Traverse Area
Surveying - Traverse
Calculating Traverse Area
E
B
A
D
C
S 6° 15’ W
189.53 ft.
S 29° 38’ E
175.18 ft.
N 81° 18’ W
175.18 ft.
N 12° 24’ W
142.39 ft.
N 42° 59’ E
234.58 ft.
N
Reference Meridian
Surveying - Traverse
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
Calculating Traverse Area
Surveying - Traverse
Calculating Traverse Area
Length (ft.) Latitude Departure Latitude Departure Latitude Departure 189.53 -188.403 -20.634 0.016 0.033 -188.388 -20. 175.18 -152.268 86.617 0.015 0.030 -152.253 86. 197.78 29.916 -195.504 0.017 0.034 29.933 -195. 142.39 139.068 -30.576 0.012 0.025 139.080 -30. 234.58 171.607 159.933 0.020 0.041 171.627 159. 939.46 -0.079 -0.163 0.000 0.
Corrections Balanced
-20.601 B A
B C
D C
-195.
-30.551E D
E A
159.974 Point E is the farthest
to the west
N
Surveying - Traverse
Calculating Traverse Area
E
B
A
D
C
S 6° 15’ W
189.53 ft.
S 29° 38’ E
175.18 ft.
N 81° 18’ W
175.18 ft.
N 12° 24’ W
142.39 ft.
N 42° 59’ E
234.58 ft.
Reference Meridian
Surveying - Traverse
DMD Calculations
N
E
B
A
D
C
Reference Meridian
The meridian distance of
line EA is:
N
E
A
DMD of line EA is the
departure of line
Surveying - Traverse
DMD Calculations
The meridian distance of line AB is
equal to:
the meridian distance of EA
+ ½ the departure of line EA
+ ½ departure of AB
The DMD of line AB is twice the
meridian distance of line AB:
double meridian distance of EA
+ the departure of line EA
+ departure of AB
N
E
A
B
Meridian distance of line AB
Surveying - Traverse
DMD Calculations
The DMD of any side is equal to:
the DMD of the last side
+ the departure of the last side
+ the departure of the present
side
DMD AB = DMD EA
+ depEA + depAB
N
E
A
B
Meridian distance of line AB
Surveying - Traverse
DMD Calculations
The DMD of line AB is departure of line AB
Side Latitude Departure
DMD
AB -188.388 -20.
BC -152.253 86.
CD 29.933^ -195.
DE 139.080 -30.
EA 171.627 159.
Balanced
Surveying - Traverse
Traverse Area - Double Area
Side Latitude Departure DMD Double Areas AB -188.388 -20.601 -20. BC -152.253^ 86.648^ 45. CD 29.933 -195.470 -63. DE 139.080 -30.551 -289. EA 171.627 159.974 -159.
Balanced
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
The double area for line BC equals DMD of line BC times
the latitude of line BC
Surveying - Traverse
Traverse Area - Double Area
Side Latitude Departure DMD Double Areas AB -188.388 -20.601 -20. BC -152.253^ 86.648^ 45. CD 29.933 -195.470 -63. DE 139.080 -30.551 -289. EA 171.627 159.974 -159.
Balanced
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
The double area for line CD equals DMD of line CD times
the latitude of line CD
Surveying - Traverse
Traverse Area - Double Area
Side Latitude Departure DMD Double Areas AB -188.388 -20.601 -20. BC -152.253 86.648 45. CD 29.933 -195.470 -63. DE 139.080 -30.551 -289. EA 171.627^ 159.974^ -159.
Balanced
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
The double area for line DE equals DMD of line DE times
the latitude of line DE
Surveying - Traverse
Traverse Area - Double Area
Side Latitude Departure DMD Double Areas AB -188.388 -20.601 -20. BC -152.253 86.648 45. CD 29.933 -195.470 -63. DE 139.080 -30.551 -289. EA 171.627^ 159.974^ -159.
Balanced
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
The double area for line EA equals DMD of line EA times
the latitude of line EA
Side Latitude Departure DMD Double Areas AB -188.388^ -20.601^ -20.601 3, BC -152.253 86.648 45.447 -6, CD 29.933 -195.470 -63.375 -1, DE 139.080^ -30.551^ -289.397 -40, EA 171.627 159.974 -159.974 -27, 2 Area = -72,
36,320 ft.^2
0.834 acre
Area =
Balanced
Surveying - Traverse
Traverse Area - Double Area
1 acre = 43,560 ft.^2
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
Side Latitude Departure DMD Double Areas AB -188.388^ -20.601^ -20.601 3, BC -152.253 86.648 45.447 -6, CD 29.933 -195.470 -63.375 -1, DE 139.080^ -30.551^ -289.397 -40, EA 171.627 159.974 -159.974 -27, 2 Area = -72,
36,320 ft.^2
0.834 acre
Area =
Balanced
Surveying - Traverse
Traverse Area - Double Area
1 acre = 43,560 ft.^2
The sum of the products of each points DMD and latitude
equal twice the area, or the double area
Surveying - Traverse
Traverse Area - Double Area
The acre was selected as approximately the amount of
land tillable by one man behind an ox in one day.
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 "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 - Double Area
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.
The word "furlong" itself derives from the fact that it is one
furrow long.
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 - 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
Side Latitude Departure DMD Double Areas AB 600.0^ 200. BC 100.0 400. CD 0.0 100. DE -400.0 -300. EA -300.0 -400. 2 Area = ft. 2 acre
Area =
Balanced
1 acre = 43,560 ft.^2
Find the area enclosed by the following traverse
Surveying - Traverse
DPD Calculations
The same procedure used for DMD can be used the
double parallel distances (DPD) are multiplied by the
balanced departures
The parallel distance of a line is the distance from the
midpoint of the line to the reference parallel or east–west
line
Surveying - Traverse
Rectangular Coordinates
Rectangular coordinates are the convenient method
available for describing the horizontal position of survey
points
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
Surveying - Traverse
Group Example Problem 5
Compute the x and y coordinates from the following
balanced.
Side Length (ft.) Latitude Departure Latitude Departure Points x y degree m inutes AB S 6 15 W 189.53 -188.403 -20.634 -188.388 -20.601 A 100.000 100. BC S 29 38 E 175.18 -152.268 86.617 -152.253 86.648 B CD N 81 18 W 197.78 29.916 -195.504 29.933 -195.470 C DE N 12 24 W 142.39 139.068 -30.576 139.080 -30.551 D EA N 42 59 E 234.58 171.607^ 159.933^ 171.627^ 159.974^ E 939.46 -0.079 -0.163 0.000 0.000 A
Bearing
Balanced Coordinates
Surveying - Traverse
Area Computed by Coordinates
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
x
y
E
B
A
D C
(159.974, 340.640)
(139.373, 152.253)
(226.020, 0.0)
(30.551, 29.933)
(0.0, 169.013)
Area Computed by Coordinates
Twice the area equals:
= 72,640.433 ft.^2
Area = 0.853 acre Area = 36,320.2 ft.^2
Surveying - Traverse
x
y
E
B
A
D C
(159.974, 340.640)
(139.373, 152.253)
(226.020, 0.0)
(30.551, 29.933)
(0.0, 169.013)
Area Computed by Coordinates
There is a simple variation of the coordinate
method for area computation
1 1
x y
1 1
x y
2 2
x y
3 3
x y
4 4
x y
5 5
x y
Twice the area equals:
= x 1 y 2 + x 2 y 3 + x 3 y 4 + x 4 y 5 + x 5 y 1
- x 2 y 1 – x 3 y 2 – x 4 y 3 – x 5 y 4 – x 1 y 5
Surveying - Traverse
x
y
E
B
A
D C
(159.974, 340.640)
(139.373, 152.253)
(226.020, 0.0)
(30.551, 29.933)
(0.0, 169.013)
Area Computed by Coordinates
There is a simple variation of the coordinate
method for area computation
Twice the area equals:
= -72,640 ft. 2
Area = 36,320 ft.^2
Any Questions?
End of Surveying - Traverse
