Surveying for Petroleum Engineers
19PTRL05CO
Lecture 5:
Theodolite :Hz and Vl Angles
Mohamed Elwageeh
Theodolite
Basic Definition
The theodolite is an instrument that is used to
determine horizontal and vertical angles through
horizontal and vertical circle readings
It is used primarily for the measurement of
angular values in a traverse or network between
survey marks, and are also used for defining
directions of lines of sight.
The main components of a theodolite are
illustrated in Figure
Theodolite Components
 Tripod: The purpose of the tripod is to provide support for the
instrument. Tripods may be telescopic, that is they have sliding legs,
or may have legs of fixed length.
 Tribrach: The tribrach is the body of the instrument carrying ass the
other parts.
 Leveling arrangement: To enable the tribrach to be leveled,
Levelling screws are fitted between the tribrach and trivet stage.
 Horizontal circle (lower plate): The horizontal circle is in reality
an accurately machined protractor graduated in a clockwise direction
and numbered from 0 to 360. The circle is therefore free to rotate
either within or around the tribrach and can be stopped in any
position by applying the lower plate clamp.
Theodolite Components
 Alidade: The alidade is the remainder of the theodolite
comprising the uprights (A frame of older instruments) which
support the telescope and vertical circle and the spirit levels.
 Controls for measuring horizontal angles:
(a) Double center system
(b) Circle setting screw
(c) Repetition clamp system
 Index marks: In order to read the circle for any pointing of
the telescope it is convenient to imagine an index mark
mounted on the alidade directly below the telescope.
 Theodolite axis or turning axis: The theodolite axis rests
on the limbs of the standards and is securely held in position
by a lockout.
Theodolite Components
 Altitude spirit level: Angles measured in vertical plane must be
measured relative to a truly horizontal line.
 Centering motion: Since the theodolite must be placed exactly
over a survey station it is fitted with a centering motion fitted usually
above the tribrach, which allows the whole of the instrument above
the tribrach to move relative to the later.
The Telescope
Reticule
Tribach & Optical Plummet
Adjusting the Horizontality
Plate (tube) Level and Levelling Screws
Adjustment of the Theodolite
The following figure shows the geometric features of
both the theodolite and total station. The most
important relationships are as follows:
(1)The axis of the plate bubble should be in a plane
perpendicular to the vertical axis.
(2) The vertical cross hair should be perpendicular to
the horizontal axis (tilting axis).
(3) The line of sight should be perpendicular to the
horizontal axis.
(4) The horizontal axis should be perpendicular to the
vertical axis (standards adjustment).
In addition, the following secondary features must be
considered:
(5) The vertical circle should be read 90o or 270o when
the telescope is level.
Theodolite Axes
Other Functions of Theodolite
Primary function of the theodolite is the
accurate measurement or layout of
horizontal and vertical angles.
Other functions:
 Determining horizontal and vertical distances
by stadia
 Extending straight lines
 Differential leveling
Setting
SettingUp
UpProcedures
Procedure
Theodolite
1
2
Setting the
tripod
Centering
using Optical
Plummet
4
3
Horizontality
using circular
bubble tube
Ensure Horizontality
using plate level
bubble tube
Dr Ahmed Ragheb
5
Focusing,
Pointing,
Reading
CIVL04C04 – Geometrics in Surveying
8
Taking Measurements
When exactly set over a survey mark and properly leveled
the theodolite can be used in two positions namely, Face left
and Face right.
The instrument is said to be facing left when the vertical
circle is on the observer’s left as he sights an object. In order
to sight the same object on face right the observer must turn
the instrument horizontally through 180o until the eyepiece
is approximately pointing to the target.
He then rotates the telescope about the transit axis thus
making the objective and of the telescope face the target the
vertical circle will now be found to be on the observer’s right.
This operation is known as transiting the telescope.
Concept
of measuring Horizontal
AngleAngle
Theodolite
Concept
of Measuring
Horizontal
Spatial Directions are projected on the horizontal circle of the
theodolite, and so Horizontal Circle Readings (HCR) are obtained
A
C
HCRA
θ
0
B
<ABC
HCRC

Hz circle fixed and pointer
moves with alidade
o
If θ –ve, then add 360
Horizontal Circle
Hz angle: It is the angle between the projection of two lines on the
horizontal plane passing by the theodolite clockwise direction
Horizontal angle <ABC = θ = HCRC – HCRA
What is important is the difference between readings not the reading itself
Dr Ahmed Ragheb
Lecture 6
CIVL04C04 - Geometrics in Surveying
6
Concept
of measuring Vertical
AngleAngle
Theodolite
Concept
of Measuring
Vertical
Face Left (FL): VL circle left of observer
Face Right (FR): VL circle right of observer
VL
h
Vertical angle (V)
hi
V = 90 – Z
Hz Plane
+ve Elevation Angle
-ve Depression Angle
It is the angle measured in the vertical plane between the line joining the
theodolite-target (line of sight) and the horizontal plane passing by the theodolite
Dr Ahmed Ragheb
Lecture 6
CIVL04C04 - Geometrics in Surveying
7
Concept of Measuring Vertical Angle
It should be remembered that the construction of
the theodolite is such that the vertical circle moves
with the telescope and the vernier or index marker
(pointer) remain fixed.
The vertical angle is measured from the line
through the index mark or arrows of the vernier
this line is made horizontal by centering the bubble
of the altitude spirit level.
It follows therefore that the bubble axis should be
parallel to the line through the vernier index marks.
Theodolite
Concept
of Measuring Vertical Angle
180
FL
o
Turn alidade 180 and telescope 180
o
0
FR
Index
270
270
90
90
A
A
180
0
V
VCRL
V
V
VR = VCRR – 270
VL = 90 – VCRL
Dr Ahmed Ragheb
VCRR
Pointer or
index fixed
and VL circle
moves with
telescope
Lecture 6
CIVL04C04 - Geometrics in Surveying
8
Index
Error
Index Error
Theodolite
180
0
FL
FR
270
δ
90
270
90
δ
A
A
0
180
V
VL = 90 – VCRL – δ
Dr Ahmed Ragheb
δ
δ
VCRL
Lecture 6
VCRR
V
VR = VCRR – 270 + δ
CIVL04C04 - Geometrics in Surveying
9
TheodoliteHow
How
determine Index
totodetermine
IndexError?
Error
Since VL = VR (for the same point)
90 – VCRL – δ = VCRR – 270 + δ
2δ = 270 + 90 – VCRL – VCRR
If δ = –ve, then index inclined in opposite
direction, substitute normally with –δ
Index error (δ)= (360 – [VCRL+VCRR])
2
If (VCRL+VCRR)≠360, then index error exists
Dr Ahmed Ragheb
Lecture 6
CIVL04C04 - Geometrics in Surveying
10
Hz/VL Angles Measurement
Theodolite
Single Angle
Angle Measurement
Measurement
Could be done with only two directions
Occup. Target
Station Station
A
HCR
FL
VCR
FR
FL
FR
B
174˚13' 02" 354˚13' 42" 88˚44' 33" 271˚16' 05"
C
12˚45' 49" 192˚46' 25" 91˚37' 19"
-----------
or more than two different directions
Occup. Target
Station Station
B
A
C
D
Dr Ahmed Ragheb
HCR
FL
VCR
FR
FL
FR
174˚13' 02" 354˚13' 42" 88˚44' 33" 271˚16' 05"
----------195˚45' 49" 15˚46' 25" 91˚37' 19"
244˚13' 30" 64˚13' 58"
-----------
-----------
CIVL04C04 – Geometrics in Surveying
4
ClosingHorizon
HorizonMeasurement
Measurement
TheodoliteClosing
Ending at the start point…….checking exists…..more accurate
Given the following tabulated theodolite readings, calculate:
i- corrected horizontal angles BAC and CAB B
C
ii- index error of theodolite
A
iii- corrected vertical angles to points B and C
Occup. Target
Station Station
A
HCR
FL
VCR
FR
FL
FR
B
C
174˚13' 02" 354˚13' 42" 88˚44' 33" 271˚16' 05"
----------12˚45' 49" 192˚46' 25" 91˚37' 19"
B
174˚13' 30" 354˚13' 58"
-----------
Given
or not
-----------
WHY?
Dr Ahmed Ragheb
CIVL04C04 – Geometrics in Surveying
5
TheodoliteClosing
Horizon
Measurement
Closing
Horizon
Solution
Hz Mean Value = HCRL + HCRR – 180
2
Occ.
Target
Station Station
B
A
C
B
HCR
FL
FR
Choose either FL
or FR degrees
Mean
HCR
174˚13' 02" 354˚13' 42" 174˚13' 22"
12˚45' 49" 192˚46' 25" 12˚46' 07"
174˚13' 30" 354˚13' 58" 174˚13' 44"
12˚46' 07“ - 174˚13' 22"
= -161˚27' 15"
-161˚27' 15" + 360 = 198˚32' 45"
Hz Angle
Corr.
Remarks
Hz Angle
198˚32' 45" 198˚32' 34"
BAC
Check ∑=360
161˚27' 37" 161˚27' 26"
CAB
If BAC + CAB = 360
OK If not (198˚32' 45"+161˚27' 37"=360˚0' 22 ") Then:
Hz closing error (φ)= (BAC+CAB) – 360 = 11" If φ –ve, substitute with –ve sign
2
Or depends on no. of directions
Corr. Hz angle BAC = BAC – φ =198˚32' 45" – (0˚0' 11") = 198˚32' 34"
Corr. Hz angle CAB= CAB – φ =161˚27' 37" – (0˚0' 11") = 161˚27' 26"
Dr Ahmed Ragheb
CIVL04C04 – Geometrics in Surveying
6
Theodolite
Vertical Angle
Angle Calculations
Vertical
Calculations
For Point A:
VCRL+ VCRR= 88˚44' 33" + 271˚16' 05" = 360˚00' 38"
Index error (δ)= (360 – 360˚00' 38") = -19"
2
Should be the same
when calculated
from Face Right
VB = 90 – VCRL – δ = 90 – 88˚44' 33" – (-0˚0' 19") = 1˚15' 46"
(Elevation)
VC = 90 – VCRL – δ = 90 – 91˚37' 19" – (-0˚0' 19") = -1˚37' 00"
(Depression)
Note: If more than one index error (VCRL+ VCRR for more than one point),
then calculate mean index error
Do not use different index error for each point
Dr Ahmed Ragheb
CIVL04C04 – Geometrics in Surveying
7
Errors in the Theodolite Angles
Instrumental Errors
(1) Error due to eccentricity of inner and outer arms.
(2) Error due to line of collimation not being perpendicular to
the trunnion axis.
(3) Trunnion axis not perpendicular to the vertical axis.
(4) Vertical axis is not truly vertical.
(5) Vertical circle index error.
(6) Error due to imperfect graduations on horizontal scale.
Errors in the Theodolite Angles
Personal Errors
(1)
(2)
(3)
(4)
(5)
(6)
In accurate centering
Error of pointing
Misreading
Improper focusing
Level bubble not centered
Displacement of tripod
Errors in the Theodolite Angles
Other Sources of Errors
(1) Poor visibility resulting from rain, snowfall or blowing dust.
(2) Sudden temperature change causing unequal expansion of
various components of a theodolite leading to errors. The
bubble is drawn towards the heated end of the theodolite.
(3) Unequal refraction causing shimmering of the signals making
accurate sighting difficult.
(4) Settlement of tripod feet on hot pavement or soft or soggy
ground.
(5) Gusty or high velocity winds that vibrate or displace an
instrument, move plumb bob strings and make sighting
procedures difficult.
m_elwageeh@hotmail.com