Angles Measurement using theodolites in geomatics engineering

MMJ14203
GEOMATIC
ENGINEERING
MRS SITI KAMARIAH MD SA’AT
FTKM, UniMAP

ANGLES MEASUREMENT
Accurate angle measurement using theodolites is crucial for precise surveying and construction in
today's engineering projects.

TOPIC OUTCOMES
 Able to compare between bearing and azimuth
 Able to compute the bearing, azimuth, and interior angles.
 Describe the field procedures that are used to set up and
measure angles with a theodolite and total station
 Booking and calculate the horizontal and vertical angles from
theodolite readings
 Recognise that the methods for setting up and measuring
angles with a theodolite are subject to many sources of error
and realize that these can be controlled provided the correct
field procedures are used

INTRODUCTION
 An angle is defined as the difference in direction between two convergent lines.

HORIZONTAL AND VERTICAL ANGLES
A horizontal angle is formed by
the directions to two objects in a
horizontal plane. Used to
determine bearings and
direction.
Vertical angles are used when
determining the height of points by
trigonometrical method. Used to
calculate slope correction for
horizontal distance

DEFINITION
 A vertical angle is formed by two
intersecting lines in a vertical plane,
one of these lines horizontal.
 A zenith angle is the complementary
angle to the vertical angle and is
directly above the obeserver
 A Nadir angle is below the observer

BASIC REQUIREMENT
To determine an angles, there are three basic requirement:
 Reference/Starting line
 Direction of Turning
 Angular of Distance
(Value of Angle)

REFERENCE LINE
 Horizontal Line
 Vertical Line

MERIDIANS
 A line on the mean surface of the earth
joining north and south poles is called
meridian.
Note:
Geographic meridians are fixed,
magnetic meridians vary with time and
location.
Relationship between
“true” meridian and grid
meridians
Figure 4.2

UNIT OF ANGLE MEASUREMENT
 The sexagesimal systems used in US and many other countries
= degree, minutes, seconds
 In Europe, used grad or gon.
 Computer computations, used radians.

TYPES OF ANGLES
Interior angles are measured clockwise or counter-clockwise between two
adjacent lines inside a closed polygon figure. The sum of interior angles in
any polygon must equal to (n-2)180o
, where n is the number of angles.
Exterior angles are measured clockwise or counter-clockwise between two
adjacent lines on the outside of a closed polygon figure. The sum of interior
and exterior angles at any station must total 360o
.
Deflection angles, right or left, are measured from an extension of the
preceding course and the ahead line. It must be noted when the deflection is
right (R) or left (L)

ANGLES TO THE RIGHT/LEFT
Angles to the right
 Measured clockwise from rear to the
forward station.
 A station are commonly identified by
consecutive alphabetical letters.
 Eg. A-B-C-D-E-F-A
 Thus the interior angles also angles
to the right.
Angles to the left
 Measured counterclockwise from rear
station.
 To avoid confusion, always observing
angles to the right.

AZIMUTH AND BEARING
 Azimuth
◦ An Azimuth is the direction of a line as given by an angle measured clockwise
(usually) from the reference meridian.
◦ Azimuth range in magnitude from 0° to 360°.
 Bearing
◦ Bearing is the direction of a line as given by the acute angle between the line
and a meridian.
◦ The bearing angle is always accompanied by letters that locate the quadrant in
which line falls (NE, NW, SE or SW).
◦ Range 0° to 90°.

COMPARISON OF AZIMUTH AND
BEARINGS

RELATIONSHIPS BETWEEN
BEARINGS AND AZIMUTHS
To convert from azimuths to bearing,
◦ a = azimuths
◦ b = bearing
Quadrant Angles Conversion
NE 0o
 90o a = b
SE 90o
 180o a = 180o
– b
SW 180o
 270o a = b +180o
NW 270o
 360o a = 360o
– b

REVERSE DIRECTION
 In figure 4.8 , the line
 AB has a bearing of N 62o
30’ E
 BA has a bearing of S 62o
30’ W
To reverse bearing: reverse the direction
Figure 4.7
Reverse Directions
Figure 4.8
Reverse Bearings
Line Bearing
AB N 62o
30’ E
BA S 62o
30’ W

REVERSE DIRECTION
 CD has an azimuths of 128o
20’
 DC has an azimuths of 308o
20’
To reverse azimuths: add 180o
Figure 4.8
Reverse Bearings
Line Azimuths
CD 128o
20’
DC 308o
20’

AZIMUTH COMPUTATION
When computations are to
proceed around the
traverse in a clockwise
direction, subtract the
interior angle from the
back azimuth of the
previous course.
When computations are to
proceed around the
traverse in a counter-
clockwise direction, add
the interior angle to the
back azimuth of the
previous course.

AZIMUTH
COMPUTATION
: CLOCKWISE
SOLUTION
START
FINISH
(Check)

AZIMUTHS
COMPUTATIO
N
Clockwise direction: subtract the interior angle from the
back azimuth of the previous course
Course Azimuths Bearing
AE 242o
55’ S 62o
55’ W
ED 314o
27’ N 45o
33’ W
DC 29o
25’ N 29o
05’ E
CB 90o
28’ S 89o
32’ E
BA 150o
00’ S 30o
00’ E

SKETCH FOR
AZIMUTH
COMPUTATION

AZIMUTHS
COMPUTATIO
N
Counterclockwise direction: add the interior angle to the
back azimuth of the previous course
COURSE AZIMUTHS BEARING
BC 270o
28’ N 89o
32’ W
CD 209o
05’ S 29o
05’ W
DE 134o
27’ S 45o
33’ E
EA 62o
55’ N 62o
55’ E
AB 330o
00’ N 30o
00’ W

BEARING COMPUTATION
Prepare a sketch
showing the two
traverse lines involved,
with the meridian
drawn through the
angle station.
On the sketch, show
the interior angle, the
bearing angle and the
required angle.

EXERCISE: BEARING COMPUTATION
 Computation can proceed in a Clockwise or counterclockwise
Figure 4.11
Sketch for Bearings
Computations

COMMENTS
ON BEARING
AND
AZIMUTHS
 Advantage of computing bearings directly
from the given data in a closed traverse, is
that the final computation provides a check
on all the problem, ensuring the correctness
of all the computed bearings

ANGLES MEASUREMENT
 All angles have three parts
 Backsight: The baseline or point used as zero angle.
 Vertex: Point where the two lines meet.
 Foresight: The second line or point

MEASURING ANGLES
 There are two methods for measuring existing or laying out new angles.
Indirect
Direct
 Indirect methods measure and lay out angles by utilizing equipment that can
not measure angles directly.
 Direct measurement and lay out of angles is accomplished by instruments
with angle scales.

ANGLE MEASURING - INDIRECT
 Tapes (or other distance measurement)
Using triangle principles
Using trigonometry based on slope angles

DETERMINING ANGLES – TAPING
Need to: measure 90° angle at point X
d d
Lay off distance d either side of X
X
l l
Swing equal lengths (l)
Connect point of intersection and X


A
B
C
Need to: measure angle  at point A
Measure distance AB
Measure distance AC
Measure distance BC
Compute angle 
α=co s
−1
(AC
2
+AB
2
−BC
2
2(AC)(AB) )


A
B
C
α=tan
−1
(PQ
AP )
Q
Lay off distance AP
Establish QP AP
Measure distance
QP
Compute angle 
P


A
B
C
sin(0.5α)=
DE
2(AD)
D
Lay off distance AD
Lay off distance AE = AD
Measure distance DE
Compute angle 
E

ANGLE MEASURING EQUIPMENT -
DIRECT
 Direct methods of measuring angles involve surveying equipment with angle
scales.
 The operator must understand how to use each type of instrument.
 Examples of Instruments:
 Sextants
 Compass
 Digital theodolites and;
 Total stations

SURVEYING COMPASS
 Previously, extensively used by surveyor.
 Temporary adjustment
 Centering: The tripod is placed with suitable height and compass fixed on
the tripod. The compass then centered over the station. Use the plumb bob
hang in center of compass.
 Levelling: The compass is leveled by the two-plate level and the levelling is
achieved by adjusting the bubbles become central in both plate level.
 Focusing the prism: The adjustment is done by prismatic compass until the
figures and graduation seen clearly

MEASURING ANGLES USING
SURVEYING COMPASS
 Read as bearing with considering N
and S, and E and W.
 The reading in degrees and minutes

THEODOLITES
General Background:
 Theodolites are surveying instruments designed to precisely measure horizontal and
vertical angles.
 They are used to establish straight and curved lines.
 To establish or measure distance (Stadia)
 To establish Elevation when used as a level (When we set the vertical angle to 90°).
They have:
 3 screw level base
 Glass horizontal and vertical circles, read directly or through a micrometer.
 Right-angle prism (optical plummet)
 High precision

THEODOLITES
Electronic read out 1” eliminate mistakes and reading the angles.
Precision varies from 0.5” – 20”
Zero is set by a button.
Repeated angle averaging.
Replacing optical theodolites (It is less expensive to purchase and
maintain).

SETTING UP THE THEODOLITES
Setting up a theodolite:
 1. Centering the theodolite
 Setup the tripod at the station. The tripod must level as possible
 Place the theodolite on the tripod. Hold the theodolite from the base and attached to the tripod head. Let the position be loose
so theodolite still can slide around tripod head.
 The ground marks is observed through optical plummet and adjusting using three foot screw.
 Fully tighten the centering screw. Look through the optical plummet again and adjust the theodolite foot screws for alignment
with the reference mark.
 2. Levelling the theodolite
 The circular bubble on tribranch is centered by adjusting the length of tripod leg
 The theodolite is rotated until the plate level is parallel to the line of any two foot screw, and adjusting the foot screw until the
plate bubble is brought to center.
 3. Elimination of parallax
 Adjusting the telescope focus on eyepiece and the object.
 the cross-hair appear clearly visible
 The image appears clear and sharp

FACE RIGHT VS FACE LEFT
 FACE RIGHT
 When the vertical circle of theodolite is on
the right of the observer, the position is
called face right and the observation
made is called face right observation
 FACE LEFT
 When the vertical circle of a theodolites
is on the left of observer, the position is
called face left and the observation is
called face left observation
By taking the mean of both face readings,
the collimation error is eliminated.

MEASURING HORIZONTAL ANGLES
Measuring Horizontal Angles:
 When theodolite is setup, point to backsight (BS).
 Zero the instrument. Write actual zero into field book
(might be not exact zero).
 Free the motion and point to foresight (FS).
 Tighten the motion and use screw for fine
adjustment.
 Read the angle, write into field book.

MEASURING HORIZONTAL ANGLES
USING THEODOLITES
We have set up the theodolite at Y, and we want the angle XYZ.
We need a target at each of X and Z, preferably the station itself, but that's not always possible.
1. Sight station X exactly, face left.
2. Set zero electronically on keyboard and book reading.
3. Change to face right, sight X exactly.
4. Book reading ( approx. 1800
)
5. Sight Z exactly.
6. Book reading.
7. Change face to face left and sight Z exactly.
8. Book reading (approx. 180o
different from <6.>
9. Extensions of this procedure may be used to measure multiple angles from a station.

MEASURING HORIZONTAL ANGLES BY
REPETITION (DIGITAL THEODOLITE)
Station B BS to A FS to C Angle ABC
Reading from left 0° 00' 00" 33° 27' 15" 33° 27' 15"
Reading from right 180° 00' 15" 213° 27' 20" 33° 27' 05"
Average 33° 27' 10"
• More reliable reading of angle value is made by repeating the
measurement. The first angle is taken from BS to FS and written into field
book.
• Then telescope is rotated against trunnion axis and pointed to FS.
• Measurement back to BS is made again then and the result is taken as an
average of both angles taken.

VERTICAL ANGLES MEASUREMENT
 Vertical angle is taken either from zenith (position at 0°) or horizon (position at 90°).
Depends on the instrument if the angle of elevation or depression has to be converted
manually (from zenith angle) or not.
 For accurate work, it is best to measure a vertical angle at least twice: once direct,
once reversed and average the result.

MEASURING ANGLES USING THEODOLITES
Observation procedures
1. Setup the instrument at B.
2. First reading in the face left position. Set the first reading as 00o
00’00” from the
reference station, and see the angles to first station.
3. Move the theodolite and read the first reference station and read as face right.
4. Complete the surveying works in loop
 Booking and calculating angles
 For horizontal angles
 The mean horizontal reading are obtain by averaging each pair of face left and face right
 For vertical angles
 FL vertical angle = 90o
–zenith reading
 FR vertical angle = FR zenith reading – 270o

EXAMPLE OF ANGLES BOOKING
RO- reference object

TOTAL STATIONS
Combined measurements
Digital display

Measures and Records:
Horizontal Angles
Vertical Angles
and
Slope Distances
Calculates:
Horizontal Distance
Vertical Distance
Azimuths of Lines
X,Y,Z Coordinates
Layout
Etc.
Total Station

ANGLE MISCLOSURE
 The sum of interior angles of a closed polygon should be:
Σ = (n-2) 180o
where n the number of sides.
 The sum of exterior angles will be:
Σ = (n+2) 180o
 Permissible misclosure can be computed by the formula:
c = K√n or K = c/ √n
where K = constant*
According to FGSC standard, K = 1.7”, 3”,4.5”,10”,12” for
1st order, 2nd Order Class I, 2nd Order Class II, 3rd Order Class I
and 3rd Order Class II

ERRORS IN ANGLE MEASUREMENT
Gross – reading, pointing, setting up over the wrong point, booking
Random – settling of tripod, wind, temperature, refraction
Systematic/instrumental
 Horizontal axis not perpendicular to the vertical axis
 Axis of sight not perpendicular to the horizontal axis
 Axis of the plate bubble not perpendicular to the vertical axis.
 Vertical index error

MISTAKES IN USING AZIMUTH AND
BEARING
Confusing magnetic and other reference bearings
Missing clockwise and counterclockwise angles
Calculation mistakes
Fail to adjust traverse angles before computing bearing or azimuth

OBSERVATION SEQUENCE
7
7
.
1
9
99.92
6
0
.
6
3
129.76
3
2
.
2
0
A
B
C
D
E
2
0
5
o
2
1
o
232
o
56
o
3
5
2
o
1
6
8
o
48
o
232
o
3
0
3
o
1
1
8
o
Start at A
End at A

CALCULATE INTERNAL ANGLES
Point
Foresight
Bearing
Backsight
Bearing
Internal
Angle
Adjusted
Angle
A
21o 118o
97o
B
56o 205o
149o
C
168o 232o
64o
D
232o 352o
120o
E
303o 48o
105o
 =(n-2)*180
Misclose
Adjustment
At each point :
• Measure foresight bearing
• Measure backsight bearing
• Calculate internal angle (back-fore)
For example, at B :
• Bearing to C = 56o
• Bearing to A = 205o
• Angle at B = 205o
- 56o
= 149o

CALCULATE ANGULAR MISCLOSE
Point
Foresight
Bearing
Backsight
Bearing
Internal
Angle
Adjusted
Angle
A
21o 118o
97o
B
56o 205o
149o
C
168o 232o
64o
D
232o 352o
120o
E
303o 48o
105o
 =(n-2)*180 535o
Misclose 5o
Adjustment o

CALCULATE ADJUSTED ANGLES
Point
Foresight
Bearing
Backsight
Bearing
Internal
Angle
Adjusted
Angle
A
21o 118o
97o
98o
B
56o 205o
149o
150o
C
168o 232o
64o
65o
D
232o 352o
120o
121o
E
303o 48o
105o
106o
 =(n-2)*180 535o
540o
Misclose 5o
Adjustment o

COMPUTE ADJUSTED BEARINGS
 Adopt a starting bearing
 Then, working clockwise around the traverse :
 Calculate reverse bearing to backsight (forward bearing 180o
)
 Subtract (clockwise) internal adjusted angle
 Gives bearing of foresight
 For example (bearing of line BC)
 Adopt bearing of AB 23o
 Reverse bearing BA (=23o
+180o
) 203o
 Internal adjusted angle at B 150o
 Forward bearing BC (=203o
-150o
) 53o

COMPUTE ADJUSTED BEARINGS
Line
Forward
Bearing
Reverse
Bearing
Internal
Angle
AB 23o
203o
150o
BC 53o
233o
65o
CD 168o
348o
121o
DE 227o
47o
106o
EA 301o
121o
98o
AB 23o
(check)
E
2
3
o
53
o
1
6
8
o
227
o
1
2
1
o
A
B
C
D
98o

EXAMPLE:
 Given below the bearing observed in traverse survey conducted with compass
where local attraction was suspected. Find the correct bearings and included
angles.
Line Fore
bearing
Back
bearing
AB 124o
30’ 304o
30’
BC 68o
15’ 246o
00’
CD 310o
30’ 135o
15’
DA 200o
15’ 17o
45’

ANSWER
1. Corrected the bearing
2. Calculate the included angles
Line Corrected Fore
bearing
Included
angles
AB 124o
30’ At A =106o
45’
BC 68o
15’ At B= 123o
45’
CD 312o
45’ At C= 64o
30’
DA 197o
45’ At D= 65o
00’

Angles Measurement using theodolites in geomatics engineering

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