1. The Traverse المضلع Civil Engineering Students Year (1)
Second semester – Phase II Dr. Kamal M. Ahmed

2. Surveying Measurements
Surveyors, regardless of how complicated the technology, measure two quantities: angle and distances.
They do two things: map ““رفع or set-out ““توقيع
Angles are measured in horizontal or vertical planes only to produce horizontal angles and vertical angles.
Distances are measured in the horizontal, the vertical, or sloped مائلةdirections.
Our calculations are usually in a horizontal or a vertical plane for simplicity. Then, sloped مائلة values can be calculated if needed.

3. For example: maps are horizontal projections of data, distances are horizontal on a map and so are the angles.
Assume that you are given the horizontal coordinates X (E), and Y (N) of two points A and B: (20,20) and (30, 40). If you measure the horizontal angle CBA and the horizontal distance AC, found them to be: 110 and 15m, then the coordinates of C can easily be computed, here is how:
Calculate the azimuth of AB, then BC
Calculate (E, N) for BC
Calculate (E, N) for C C B A

4. But, if you were given a slope distance or a slope angle, you won’t be able to compute the location (Coordinates) of C.
What we did was to map “نرفع” point C, we found out its coordinates, now you plot it on a piece of paper, a “map” is a large number of points such as C, a building is four points, and so on.
Now, if point C was a column “عمود” and we wanted to set it out “نوقعها”, then we know the coordinates of C from the map:

5. You set out a point, then you can set out a project.
Calculate the angle ABC and the length of BC
Setup the instrument “الجهاز” ,such as a theodolite, on B, aim “وجة” at A
Rotate the instrument the angle ABC, measure a distance BC, mark the point.
You set out a point, then you can set out a project.
In both cases, you need two known points such as A and B to map or set out point C
We call precisely known points such as A and B “control points”
In horizontal, we do a traverse “مضلع” to construct new control points based on given points.
You need at least two points given in horizontal ( or one and direction) and one in vertical to begin your project

6. Traversing Definition: Why?
A traverse is a series of consecutive lines whose lengths and directions have been measured.
Why?
The purpose of establishing a traverse is to extend the horizontal control. A survey usually begins with one given vertical control and two ( or one and direction) given in horizontal
You need more than two points to control the project, have enough known points to map any point, and set-out any object any where in a large project. 2

7. خطوات عمل المضلع Procedure
Grass
N (mag) A C D E B
Assume that you wanted to map “calculate coordinates of the building, trees, and the fence “سور” in the drawing, you are given points A and B only, cannot measure angle and distance to corner F or the trees!! F
You need to construct new control points “points of known precise coordinates” such as C, D, and E to measure from.
You do that with a traverse

8. خطوات عمل المضلع Procedure
Grass
N (mag) A C D E B
Walk around and decide which are the best locations to have new control points
Construct the points, nails on asphalt, concrete and bolts, etc.
Measure all the angles and all the lengths of the traverse
Check if the angles and lengths are accepted
If rejected, re-do the work
If accepted, adjust the errors and compute coordinates.

9. Coordinate Computations
Assume that we were given a site to map, and the coordinates of one point (A), and the azimuth of the line (AB), we need more known (control stations)
We marked three more points around the site, the four points make rectangle (or a square).
We then measured all the internal angles and the length of all the sides (lines).
Using the given azimuth of AB and all angles, we computed the azimuth of all the sides, we get the following table:

10. Coordinate Computations
Point
Line
Length
Azimuth
(a)
DE =
d sin( a)
DN =
d cos( a) E N A
200.00
350.00 AB
100.10
0° 00' 00''
 0.00
 100.10 B
450.10 BC
100.00
90° 00' 00"
100.00 
0.00  C
300.00 CD
180°00'00"   D
350.10 DA
99.70
270°00'00”
- 99.70
200.30
Sum
399.80
 0.30
 0.10

11. Questions
Note that the coordinates of A when computed at the bottom of the table, are not the same as given coordinates.
Also note the relationship between that error and the sum of Northings and Eastings.
How do you explain that?
Assume that the traverse was a perfect square of 100 m side length and oriented towards the north, what you notice in our measurements, and how can you relate that to the error in A?

12. The concept of Linear Closing ErrorB c D
Closing error = 0.32m
Notice that if the corrections are ignored, the value of the errors will
Appear when you re-compute the coordinates of the first point (A).
ΔE=0.3
ΔN=0.1

13. The concept of Angular Closing Error
Now let us assume the same perfect square of 90° angles and 100 m sides. When measured all lengths were correct, while the angle at B was in error by 10° when measured, the surveyor reported 100°. What happens? A B c D
Here is the perfect traverse that we are trying to measure:

14. The concept of Closing Error
Error in angles OR error in distances will result in a closing error: last point will not be at the first point.
The problem is that we do not know where the errors are and how much each error?
Measurements are never exact, we always assume that we have errors in angles and distances.
Before we learn how to compute the errors and how to adjust for them, let us learn some issues with traverse

15. Closed and Open Traverses
A closed traverse is the one that starts and ends at known points and directions, whether the shape is closed or not
A closed traverse can be a polygon {closed shape} or Link {closed geometry-open shape

16. Closed (polygon or link) traverses

17. Open Traverses are not used in engineering control applications, why?
True Location
Observed Location
∆X=XB-XA A L1 L2 L3 L4
∆Y=YB-YA L5
∑∆YObserved B MX MY
MXY
∑∆XObserved
Open Traverses are not used in engineering control applications, why?
The problem: there is no way to check the for the errors; you will have to accept whatever coordinates computed.

18. Traverse Notations
We will only cover the closed Traverse with interior angles measured. 3

19. Traverse Stations Successive stations should be inter visible.
Stations are chosen in safe, easy to access places.
Lines should be as long as possible
To reduce the number of lines
Short lines will produce less accurate angles, the traverse gets distorted as shown below. A B T1 T2 T3 T4 5

20. Traverse Stations
Angles should be as equal as possible and better be 30 to 150°, why????
Lines should be and as equal as possible, Why?
Stations must be referenced to retrieve them if lost. We produce a “descriptive card” for each point
Descriptive card for a traverse point

21. شكل (8.1) كروكي عام للمنطقةأ ب
مبني سكني وأداري
طريق ب
طريق أ
طريق رئيسي ج د
هـأ
الشمال
روبير (110.25م)
حديقة
شكل (8.1) كروكي عام للمنطقة
3. عمل كروكيات النقط
سلم مبني
عمود أناره
شجرة
11.24 م
8.67 م
12.73 م
شكل (8.2) كروكي (كرت وصف) نقطة ج

22. تثبيت نقط المضلع :
تثبت نقط المضلع أما بوتد خشبي أو زاوية حديدية في الأراضي الزراعية
أو مسمار حديد في المناطق الصخرية والإسفلتية أ
و بقاعدة خرسانية وزاوية حديدية في المناطق الرملية
ويستحسن وضع علامة بالبوية كلما أمكن على النقط يوضح رقمها وتاريخ تثبيتها .

25. Traversing by Interior Angles
All internal angles and all horizontal distances are measured
Each angle is measured in direct and reverse,
Each angle is observed at least three times.
A line of known direction should either be given or assumed, what is a line with known direction?
If the line of known direction is not a member of the traverse, the angle to a traverse member should be measured. Why? 2

26. Closing Error In Traverses
We measure two values: angles and distances
Because of errors in both measurements, we get angle misclosure and linear misclosure ( closing error).
Both types of errors result in error in closure, we need a way to separate the error of angles from the error in distances to check and adjust them separately

27. The concept of Angle Misclosure
Here is how the measured traverse will look:
Line AB was correct A B c D
Line BC was correct, but angle A was wrong
The rest of the lines and angles are correct A’

28. Computations and Adjustments of Angle Misclosure
The sum of internal angles of a polygon of (n) points =
(n - 2) * 180o
Angle misclosure = difference between the sum of the measured angles and the geometrically correct total for the polygon.
The misclosure is divided equally among the readings keeping in mind the measuring accuracy, and should be done at the beginning of the adjustment. 3

29. Judging The Angle Misclosure
Usually the standards give an equation of the form:
Max allowed angle misclosure c = k * n where (n) is the number of points and K is a constant defined according to which standards used
If angles are accepted, correct by dividing the error equally among the angles
For example: The Federal Geodetic Control Subcommittee: 1.7, 3, 4.5, 10, and 12” for first-order, second-order class I, second-order class II, third-order class I, third-order class II

30. If no standards were given, then assume that:
Allowed angle misclosure = 3n =
3 * theodolite accuracy *  No. of angles
Example (1)
For 5 points traverse, the sum of internal angles is
538° 40’ and the accuracy of the measured angles is 10’. Do you accept the observations?.
Answer
Ang. Clos. Err. = – 180 x (5 – 2)
= – 540 = = - 80’
Allowable Angle Closing Error = 3 x 10’ √5 = 67’
Reject the observations. You have to re-observe

31. Example (2)
If the angles were observed to the nearest 30” in 5 points traverse. If the angular closing error was 2’, correct the angles.
Answer
Allowable angular closing error = 3 x 30” √ 5 = 201”
As Δθ = 120” (≤ 201” ) , accepted.
Correction in each angle = “ / 5 = ”
But the angles were observed to the nearest 30”, the corrections will be :
- 30”, - 30”, - 30” , - 30” & 0 .

32. The concept of Linear Closing ErrorB c D
Assume that the traverse in reallity was a perfect square.
Assume that there was an error in measuring the length AB only, all other lengths and angles were correct A’
ΔE
ΔN
- A will close at A’,
- AA’ is the linear closing error

33. If the traverse is closed, then ΔE = 0 and ΔN = 0
EDA
ECD N
- ve
- ve D C
EBC A
EAB B E
If the traverse is closed, then
ΔE = and
ΔN = 0
+ ve
+ ve

34. If the traverse is closed, then ΔE = 0 and ΔN = 0
EDA
ECD N
- ve
- ve D
ΔN C
EBC
ΔE
If the traverse is closed, then
ΔE = and
ΔN = 0 A’ A
EAB B E
If the traverse is not closed,
+ ve
+ ve
Then ΔE = Ec and ΔN = Nc

35. Computations of Linear Closing Error
If he closing error is (W) then
Ew = ΔE and
Nw = ΔN,
W = length of closing error =  Ew2 + Nw2
Fractional Closing error = traverse precision = W /  L
Direction of the error = Azimuth =
tan-1 (Ew / Nw) = tan-1 (ΔE / ΔN)

36. Adjustment of Linear Misclosure
Compute and adjust the angle misclosure
Compute the linear misclosure:
Compute the azimuth of a traverse side
Compute the azimuth of all the sides
Compute the departure and latitude of all the sides
Compute the Misclosure in (E) direction =
sum of the departures.
Compute the Misclosure in (N) direction =
sum of the latitudes.
Compute the linear misclosure
If accepted, use the Compass (Bowditch) rule to adjust: 4

37. Compass (Bowditch) Rule
Correction in departure for AB = - (
 ΔE
 L
) ( LAB)
Correction in latitude for AB = - (
 ΔN
 L
) ( LAB)
Where:
L is the length of a line, and ( L) is the perimeter.
- Advantages and disadvantages of this method

38. Computations of Coordinates
Add the corrections to the departure or the latitude of each line to get the adjusted departure or latitude (maintain signs)
Compute the adjusted point coordinates using the corrected departure or latitude:
Ei = E i-1 + ΔE
Ni = N i-1 + ΔN
Check that the misclosure is zero.

39. Example of Standards
Horizontal Control Accuracy Standards For Traverse
(By The Federal Geodetic Control Subcommittee (FGCS))
3 rd
2 nd
1st
Order II I
Class
12.0”√n
10.0”√n
4.5”√n
3.0”√n
1.7”√n
Angular Closure
0.80√∑L
or,
1/5,000
0.40√∑L
1/10,000
0.20√∑L
1/20,000
0.08√∑L
1/50,000
0.04√∑L
1/100,000
Linear Closure
(after angul. adj.)

40. Acceptable Relative precision
1 in & 30” √n for most engineering surveys
1 in & 10” √n for control, for large projects
1 in & 2” √n for major works and monitoring for
structural deformation etc.
Ex.: The angles were observed to the nearest 30” in 5 points traverse. If the angular closing error was 2’, correct the angles.
Answer
Allowable angular closing error = 3 x 30” √ 5 = 201”
As Δθ = 120” (≤ 201” ) , accepted.
Correction in each angle = “ / 5 = ”
But the angles were observed to the nearest 30”, the corrections will be :
- 30”, - 30”, - 30” , - 30” & 0 .

41. point
Length L
Azimuth AZ
Departure
L sin (Az)
Latitude
L cos (Az)
Correction
Balanced E N
(WN/L)* L
(WE/ L)* L A
285.10 ’
125.72
255.88 B
610.45 ’
590.77 C
720.48 ’ D
203.00 ’
-5.99
202.91
747.02 ’
388.5
Sum P=
WE =+0.54
WN =-0.72

42. =(0.54/2466.06)x285.1 =(0.72/2466.06)x285.1 N E Balanced CorrectionN E
Balanced
Correction
Latitude
= L cos (AZ)
Departure
= L sin (AZ)
Azimuth
Length
Pnt.
Lat.
Dep.
(WE/ L)* L
(WN/ L)* L AZ L A
255.96
125.66
0.08
0.06-
255.88
125.72
26 10’
285.1 B
590.64
0.18
0.13-
590.77
104 35’
610.45 C
0.21
0.16- ’
720.48 D
202.97
-6.03
0.06
0.04-
202.91
- 5.99 ’
203
388.69
0.19
0.14-
388.5
-517.4
647.02
 check
0.00
0.72
-0.54
WN=0.72
W =+0.54E
=L
Sum
=(0.54/ )x285.1
=(0.72/ )x285.1

43. Other Methods
There are several methods that are used to adjust or balance traverses;
Arbitrary method
Transit rule
Least-Squares method

44. Traverse Area C D B E A Traverse area = 1 S { Ei (Ni+1 - Ni-1)} 2
Multiply the X coordinate of each point by the difference
in Y between the following and the preceding points, half the sum
is the area
The formula will work for traverses lettered in a clockwise direction, but it will give a correct area with a negative sign.
The formula should work if you switch the N and the E. 6

45. Example
Calculate the area of a traverse whose corners are (100,100), (300,100 ), (300,300), (100,300).