Triangulation

Triangulation
N.M.A. Wijeratna
Institute of Surveying and Mapping
Diyatalawa, Sri Lanka

Triangulation
Classification and Accuracy
Design and Layout
Strength of Figure
Scale and Azimuth Controls
Stations, Signals, Beacons
Indivisibility between triangulation stations
Triangulation theodolites
Etc….

Geodetic Surveying
Geodetic Surveying – surveying
technique to determine relative positions
of widely spaced points, lengths, and
directions which require the consideration
of the size and shape of the earth.
(Takes the earth’s curvature into
account.)

Geodetic Survey :- When survey
extends over a large areas more than 200
sq. km. and degree of accuracy is also
great. The curvature of earth is also
taken into account.
Geodetic survey is used to provide
control points to which small surveys can
be connected.

SURVEY for Establishment of Controls
Horizontal positioning Vertical positioning
TRIANGULATION
TRILATERATION
TRAVERSING
ASTRONOMICAL
POSITIONING
GEODETIC LEVELLING
TRIGONOMETRIC
HEIGHTING
BAROMETRIC
LEVELLING

Horizontal Control
Established using
Traversing
Triangulation
Trilateration
Astronomical Positioning
Global Positioning System (GPS)

Triangulation
The process of a measuring system
comprised of joined or connected
triangles whose vertices are stations
marked on the surface of the earth and in
which angular observations are supported
by occasional distance and astronomical
observation is known as triangulation
Use to established a horizontal control
system for a country

Principle of Triangulation
Entire area to be surveyed is converted
into framework of triangles
If the length and bearing of one side and
three angles of a triangle are measured
precisely, the lengths and directions of
other two sides can be computed
This method of surveying was first
introduced by a Dutchman called Snell

Precisely measured line is called based line
Computed two lines are used as base lines
for two interconnected triangles
Vertices of the individual triangles
are known as triangulation stations
Extending this process network of triangles
can be computed over the entire area
As a check the length of one side of last
triangle is also measured and compared
with the computed one

subsidiary bases are measured at suitable
intervals to minimize accumulation of
errors in lengths
Astronomical observations are made at
intermediate stations to control the error in
azimuth
Those triangulation stations are called
Laplace Stations.

Purpose of Triangulation
To establish the accurate control points for
plane and geodetic surveys of large areas
To establish the accurate control points for
photogrammetric surveys
Accurate location of engineering works

Classification and Accuracy
The triangulations classified in to three
 Primary triangulation or First order triangulation
 Secondary triangulation or Second order
triangulation
 Tertiary triangulation or Third order triangulation

Primary Triangulation
Highest grade of triangulation system
To determine the shape and size of the
earth's surface or for providing precise
plainmetric control points for subsidiary
triangulations
Stations are generally selected 16 km to
150 km apart

Specifications of Primary
Triangulation
Length of the base lines - 8 km to 12 km
Length of the sides - 16 km to 150 km
Average triangular error - less than 1"
(after correcting for spherical excess)
Maximum station closure – less than 3”
Actual error of the base - 1:50000
Discrepancy between two measurements
– 5 mm Km

Secondary Triangulation
To connect two primary triangulations
to provide control points closer together
than those of primary triangulation
For the densification of horizontal
control net work

Specifications of Secondary
Triangulation
Length of the base lines - 2 km to 5 km
– 10 mm Km

Tertiary Triangulation
To provide control points between
stations of primary and second order
triangulation
For the densification of horizontal
control for topographical surveys on
various scales.

Specifications of Tertiary
Triangulation
Length of the base lines – 100 m to 500 m
– 25 mm Km

Layout of Triangulation
The arrangement of the various triangles
of a triangulation series, is known as the
layout of triangulation
Three types of layout
(1) Simple triangles in chain
(2) Braced quadrilaterals in chain
(3) Centred triangles and polygons

Simple triangles in chain
Used when control points are provided in a
narrow strip of terrain such as a valley
between two ridges.
Rapid and economical due to its simplicity
of sighting only four other stations and
observations of long diagonals are avoided.
Not provide any check on the accuracy of
observations as there is only one route
through which distances can be computed.

check base lines and astronomical
observations for azimuth at frequent
intervals are very essential to avoid
excessive accumulated error

Braced Quadrilaterals in chain
Consists of figures containing four corner
stations and observed diagonals

Braced Quadrilaterals system is treated
to be the best arrangement of triangles as
it provides a means of computing the
lengths of the sides using different
combination of the sides and angles

Centred Triangles and
Polygons
Consists of figures containing centered
polygons and centered triangles

Used when vast area extending in all
direction is required to be covered.
The centered figures generally are
quadrilaterals, pentagons, or hexagons
with central stations.
Provides proper check on the accuracy of
the work
low progress of the work due to more
settings of the instrument

Factors to be consider in
selecting a figure
Simple triangles should be preferably
equilateral
Braced quadrilaterals should be preferably
squares
Centered polygons should be regular
No angle of the figure, opposite a known
side should be small

The angles of simple triangles should not
be less than 45°
In case of quadrilaterals no angle should
be less than 30°.
In case of centered polygons, no angle
should be less than 40°
The sides of the figures should be of
comparable length

Two methods
Grid iron system
 Centered system
Layout for Primary Triangulation

Grid iron system
Primary triangulation is laid in series of
chains of triangles
Usually run roughly along the meridians
(north-south) and along the perpendiculars
to the meridians (east-west) throughout the
country
Distance between two chains may vary
from 150 km to 250 km

Area between the parallel and
perpendicular series of primary
triangulation are covered by the
secondary and tertiary triangulation
systems.
Adopted in India, Austria, Spain and
France

Centered system
Whole area of the survey is covered by a
net work of primary triangulation extending
outwards in all directions from the initial
base line
Base line is generally laid at the centre of
the country
Used for the survey of an area of moderate
extent
Adopted in United Kingdom

Well Conditioned Triangles of
a Triangulation System
The arrangement of triangles in the layout
and the magnitude of the angles in
individual triangles, affect the accuracy of
a triangulation system.
The shape of the triangle in which any
error in angular measurements, has a
minimum effect upon the lengths
of the computed sides, is known as a well
conditioned triangle.

Hence, the best shape of a triangle is an
isosceles triangle whose base angles are
56° 14' each.
But, for all practical purposes, an
equilateral triangle may be treated as a
well conditioned triangle.
Note. The triangles, whose angles are
less than 30° or more than 120°, should
be avoided in the chain of triangles.

Scale error
Scale error depends on the accuracy of
the base line measurements.
Defined as the difference between the
measured and computed check base
Control providing frequent base line
measurements

Distance Angle
Angles that involve for the calculation of
an unknown side are the Distance angle
Two distance angles involve for the
calculation of an unknown side
Namely angle opposite to the known side
and the angle opposite to unknown side

No effect by the remaining angle on the
calculation unknown side and referred to as
Non effective angle
Station should be selected such that non
effective angle is quite smaller and distance
angles are large for the accurate computation
of sides
Known
Unknown

Strength of Figures
Strength of figure can be defined as a figure
which gives the least error in calculated
length of last line in the system due to the
shape of triangles and computation of the
figures.
If it is assumed that all angles are
measured with the same degree of
precision, the strength of figure depends
upon following factors.

Number of observed angles or directions
Number of geometric conditions
Properties of each triangles in which the
size of distance angles used in
computation

Relative strength of figure
Relative strength of the figure is denoted
by R and depends on the following
factors :
 (a) Number of observed directions.
 (b) Number of trigonometric conditions.
 (c) Magnitudes of the distances and angles.

D = number of directions observed
excluding the known side of the given
figure


 d
D
CD
R
Where
22
BBAAd  

δA- Tabular difference for 1” error in
distance angle A (Difference per second
in the sixth decimal place of logarithm of
sine of the distance angle A)
δA- Same as for A but for distance angle
B
C = number of the geometric conditions
(Angles + sides)
Where
)32()1''(  snsnC

Example
Calculate the strength of the following
figure ABCD for each of the four roots by
which BD can be computed from the
known side AC. All stations were
occupied and all angles were measured

Station marks
• Object of station marks is to provide a surface
mark with a permanent mark buried below
the surface on which a target or instruments is
to centered over it
• Should be bronze or copper marks cemented
into rock or concrete surface
• Normally station marks are buried below the
ground surface to protect from the
disturbances

• Control points can be constructed in concrete
with center mark in bronze or copper and
buried at the required place
• Few dead measurements are taken using
permanent features around the station and
keep as a diagram

Satellite Station
• In order to secure well condition triangle or
better intervisibility objects such as church tops,
plag poles or towers etc. are sometime selected
as triangulation stations
• If the instruments is impossible to set up over
that point a subsidiary station known as a
satellite station or false station is selected as near
as possible to the main station
• Observations are made to the other stations with
the same precision from the satellite station

Reduction to center
• The angles are then corrected and reduced to
what they would be from the true station
• The operation applying to this correction due to
the eccentricity of the station is generally known
as reduction to center
• Distance between true station and satellite
station is determined by method of trigonometric
levelling

Signals
• They are the devices erected to define the
exact position of a station
• A signal is placed at each station so that the
line of sights are established between
triangulation stations
• Signals may be opaque (3-4 legged type) or
luminous

Classification of Signals
• (a) luminous signals
• (b) opaque signals.
• Luminous Signals. Luminous signals are
further divided into two categories
– sun signals
– night signals.

Sun Signals
• Those signals which reflect the rays of the sun
towards the station of observation, are known
as heliotropes.
• Such signals can only be used in clear weather.
• Heliotropes do not give better results as
compared to the opaque signals.

Night Signals
• While making observations at night, night
signals are used.
• Various types of night signals are:
– Various forms of oil lamps with a reflector. These
are used for sights less than 80 km.
– Acetylene lamps designed by captain G.T. McCaw.
These are used for sights more than 80 km.

Opaque Signals
• The opaque, or non-luminous signals
used during day
• Various forms and the ones most commonly
used are the following
– Pole Signal
– Target Signal
– Pole and Brush Signals
– Stone Cairn
– Beacon

Pole Signal
• It consists of a round pole painted black
and white in alternate and is supported
vertically over the station mark on a
tripod.
• Pole signals are suitable up to a distance
of 6 km.

Target Signal
• It consists of a pole carrying two square
or rectangular targets placed at right angles to
each other
• The targets are generally made of cloth
stretched on wooden frames
• Target signals are suitable up to a distance of
6 km

Pole and Brush Signals
• It consists of a straight pole about 2·5 meter long
with a bunch of long grass tied symmetrically
round the top making a cross
• The signal is erected vertically over the station
mark by heaping a pile of stones up to 1·7 meters
round the pole
• A rough coat of white wash is applied to make it
more conspicuous to be seen against a black back
ground

Stone Cairn
• It consists of stones built up to a height of 3
meters in a conical shape.
• This white washed opaque signal is very
useful if the back-ground is dark.

Beacon
• It consists of red and white cloth tied round
the three straight poles.
• This can be easily centered over the station
marks
• Beacons are useful when simultaneous
observations are made at both the stations.

Properties of an Ideal Signal
• Should be clearly visible from a distance
against any back ground.
• It should provide easy and accurate bisection
by a telescope
• It should be capable of being accurately
centred over the station mark.

Phase of the signals
• When using circular shape signals the
illumination part depends on the direction of
sunlight (Due to lateral illumination)
• Observer has a tendency to bisect the center
of the illuminated part
• This cause an angular error known as Phase
error

Phase error
• Observed angle must be reduced to the
corresponding center mark
• This effect is most common in cylindrical
signals
• Let “O” be the center of a signal at a distance
d from the observer and “O' “ be the false
center. Let r be the radius of the signal and A
is the observer’s position.

• Phase error is given by
onds
d
r
d
r
e sec206265
)2/(cos
"1sin
)2/(cos 22


O O'
Sun
d
A
α
θ
• Where α is the angle
which the direction of
sun make with OA

Tower
A tower is erected at the triangulation station
when the station or the signal or both are to
be elevated to make indivisibility between
station
The height of tower depends upon the
character of the terrain and the length of
the sight

 Towers generally have two independent structures
 Outer structure is for supporting the observer and the signal
whereas the inner one is for supporting the instrument only
 The two structures are made entirely independent of each
other so that the movement of the observer does not disturb
the instrument setting
 Two towers may be made of masonary, timber or steel
 Timber scaffolds are most commonly used, and have been
constructed to heights over 50 m

Reconnaissance
• Objective of the reconnaissance is to plan a
system of triangulation in accordance with the
specifications laid down for the type of
triangulation
• A though reconnaissance of the area contributes
to the accuracy, geographical strength, simplicity
and economy

• Reconnaissance team should prepare a well
defined description of the stations selected by
them due to two main reasons
– Observation team will identify the stations from
this report
– Stations are to be used for decades
• Description must include the approximate
directions to the prominent topographic
features. An aerial photographic identification
should be include if possible

• Method of approach and the path to be
followed to the station should also be
included
• Locations of water, camping sites and local
food supplies can also be noted

Criteria for selection of triangulation
stations
• Triangulation stations must be selected
carefully
• It can save a lot of time and funds by keeping
the following key points in mind
– Triangulation stations should be intervisible.
– Stations should be easily accessible with
instruments.
– Station should form well-conditioned triangles.

– Stations should be located so that the survey lines
are neither too small nor too long
– Small sights cause errors of bisection and centering
and Long sights cause direction error as the signals
become too indistinct for accurate bisection
– Stations should be useful for providing intersected
points and also for detail survey
– Cost of clearing and cutting and building towers
should be minimum
– No line of sight should pass over the industrial
areas to avoid irregular atmospheric refraction
– Main lines are within the area to be surveyed

Determination of intervisibility and
height of triangulation stations
• Intervisibility between stations is the most
essential condition in triangulation
• When the distance between stations is too large
or the elevation difference is less both signal
and station to be elevated to overcome the
effect of the earth curvature
• The calculations of height of signal and
instrument depends upon the following factors

– Distance between the stations
– Relative elevation of the stations
– Profile of the intervening ground
Distance between the stations:
• If the intervening ground does not obstruct
the intervisibility, the distance of horizon
from the station of known elevation is
given by the following formula

• Distance between two stations A and B of
heights hA and hB respectively is D.
• DA and DB are the distances of visible horizon
from A and B respectively and h’B is the
required elevation at B above the datum
Relative elevation of the stations

• Since DB = D – DA, h’B can be
calculated and checked whether to be
raised above the ground.
• Normally the line of sight is kept at 3m
above the ground as the refraction is
maximum near the ground

Profile of the intervening ground
• This is done by plotting peaks of the
undulating ground to ensure the propose line
of sight passing above the obstructions

• The height of the station to be raised can be
calculated using previous equation or using
Captain G.T. McCaw’s formula

• Practically in most of the cases, the zenith
distance is very nearly equal to 90° and,
therefore, the value of cosec² ξ may be taken
approximately equal to unity
      




 

R
m
ecxs
s
x
hhhhh ABAB
2
21
cos
2
1
2
1 222

• Captain G.T. McCaw’s formula

• If h > hc , the line of sight is free of
obstruction. In case , hc < h the height of
tower to raise the signal at B
      06735.0
2
1
2
1 22
 xs
s
x
hhhhh ABAB

Example
• There are two stations P and Q at elevations of
200 m and 995 m, respectively. The distance of Q
from P is 105 km. If the elevation of a peak M at
a distance of 38 km from P is 301 m, determine
whether Q is visible from P or not. If not, what
would be the height of signal required at Q so
that Q becomes visible from P ?

• If the height of line of sight is greater than 995 m
the obstruction by intervening ground
• Otherwise no obstruction
105 Km
hC = 301 m

• We should find QQ”
• QQ” = QQ’+Q’Q”
105 Km
hC = 301 m
Q’
Q
Q”
M”
M’
M
P
P’ T

• From Similar Triangle P’M’M” and P’Q’Q”
PM
MM
PQ
QQ
MP
MM
QP
QQ "'"'
''
"'
''
"'
 1
 
KmPT
hPT
KmPTPMPTMT
MTMM
MMMMMM
45.54
200853.3853.3
38
06735.0'
'""'
2





2
3• From
3
KmKmKmMT 45.163845.54 

2• From
mmmMM
MMMMMM
mMM
MM
77.28223.18301"'
'""'
23.18'
)45.16(06735.0' 2




1• From
mQQ
mQQ
PM
MM
PQ
QQ
34.781"'
77.282
38
105
"'
"'"'




• As the elevation 995 m of Q is more than 953.44 m, the
peak at M does not obstruct the line of sight.
 
 
mQQ
mmQQQQQQ
mmQQ
KmKmKmQT
PTQPQT
QTQQ
QQQQQQ
44.953"
34.78110.172"''"
10.17255.5006735.0'
55.5045.54105
06735.0'
"''"
2
2








105 Km
hC = 301 m
Q’
Q
Q0
M0
M’
M
P
P’ T
• We should find the elevation of line of sight at
peak M
• From Similar Triangle P’M’M0 and P’Q’Q0
1

• Elevation of the line of sight at M is 316.04m
but elevation of peak is 301m
mMM
mmMMMMMM
mmMM
QQ
PQ
PM
MM
mQQ
mmQQQQQQ
PM
MM
PQ
QQ
MP
MM
QP
QQ
04.316
81.29723.18''
81.29790.822
105
38
'
''
90.822'
10.172995''
''
''
'
''
'
0
00
0
00
0
00
0000








Measurement of Horizontal Angles
• Required instruments with great degree of
refinement
• Early days greater refinements was obtained by
increasing the diameter of the horizontal circle
• Later it was obtained by micrometer theodolites
• At present micrometer theodolites were replaced
by double reading theodolites with optical
micrometer

Characteristics of optical micrometer
theodolites
• Small and light
• Graduations are on glass circles and much
finer
• The mean of two readings on opposite sides is
read directly in an auxiliary eye piece generally
beside the telescope
• They are electrically illuminated
• Completely water and dust proof

Triangulation Theodolites
• There are two types of triangulation
theodolites used for precise work
–Repeating Theodolites
–Direction Theodolites

Repeating Theodolites
• Characteristic feature of repeating theodolites
is that it has a double vertical axis (Two
centers and two clamps)
• Examples:
»Watts Microptic theodolite No 1
»Ordinary transit theodolites
»Vernier theodolites

Direction Theodolites
• Direction theodolite has only one vertical axis
and a single horizontal clamp and tangent
screw which controls the rotation about
vertical axis
• Optical micrometers are used to read
fractional parts of the smallest divisions of the
graduated circle

Direction Theodolites
• There are number of direction theodolites
used for first and second order triangulations
• Examples:
– Wild T-2 theodolites
– Wild T-3 precision theodolites
– Wild T-4 Universal theodolites

Methods of Horizontal Angle
Measurements
• Two methods for observing
angles in triangulation
–Repetition method
–Direction method

• In repetition methods each angle is measured
number of times using different parts of the
circle independently with a vernier theodolite
• In direction method several angles at a station
are measured in terms of directions of their
sides from the initial station. Direction
theodolites are used
• Normally Repetition theodolites are
recommended for second and third order
triangulation while direction theodolites are
for primary triangulation

Triangulation Adjustments
• Even after exercising care and precautions
unavoidable error are with observations
• So observations should be adjusted
distributing the observational errors
• Most accurate method for adjustment is least
square adjustment but it is complicated as all
the angles are simultaneously involved

• Adjustment can be achieved adjusting angles,
stations and figures separately
• After figure adjustment sides are calculated
using sine rule
• Positions of the points are determined
calculating the geodetic coordinates

Angle adjustment
• There are number of geometrical conditions to be
satisfied in any triangulation figure
• All the conditions never meet due to errors in the
observations
• Necessary to adjust the angles to find the most
probable or best values
• Correction to be applied is directly proportional
to weight and also to the square of probable
error
• Most probable value is the arithmetic mean or
weight arithmetic mean

Station adjustment
• Determination of most probable values of the
angles at a station to satisfy geometric
consistency
• Obtained error is distributed equally if the
weights are equal otherwise error is distributed
inversely proportional to the weight
• If different angles are measured normal
equations are formed and solved simultaneously

Figure Adjustment
• In any triangulation system determination of
most probable values of the angle to fulfill the
geometric conditions is called figure
adjustment
• Best or most probable value can be obtained
by the method of least square adjustment
(Also known as Rigid method)

Adjustment of a triangle
• Three angles in triangle must be adjusted
using following rules
• Let A, B, C – Angles of a triangle
n – number of observations
w – weight of the angle
d – discrepancy (error of closure)
Co – correction to observed angle

• Rule 1: For angles of equal weight the
discrepancy is distributed equally among all
the angles
• Rule 2: For angles of unequal weight the
discrepancy is distributed inversely
proportional to the weights
dCoCoCo cBA
3
1

d
www
w
d
www
w
d
www
w
cBA
C
c
cBA
B
B
cBA
A
A CoCoCo 111
1
,
111
1
,
111
1







• Rule 3: If the number of observation is given
discrepancy is distributed inversely
proportional to number of observations
• Rule 4: Discrepancy is distributed inversely
proportional to square of the number of
observations
d
nnn
n
d
nnn
n
d
nnn
n
cBA
C
c
cBA
B
B
cBA
A
A CoCoCo 111
1
,
111
1
,
111
1






d
nnn
n
d
nnn
n
d
nnn
n
cBA
C
c
cBA
B
B
cBA
A
A CoCoCo 222
2
222
2
222
2
)1()1()1(
)1(
,
)1()1()1(
)1(
,
)1()1()1(
)1(







• Rule 5: Discrepancy is distributed proportional
to square of the probable error
• Rule 6: When the weights of the observations
are not given directly, if “v” is the difference
between the mean observed value and the
observed value of an angle the weight of the
angle is given:
d
EEE
E
d
EEE
E
d
EEE
E
CBA
C
c
CBA
B
B
CBA
A
A CoCoCo 222
2
222
2
222
2
,,







 2
2
2
1
Av
n
w

• Rules 1, 2 and 6 are sufficient to adjust the angles
2
2
2
1
1
A
A
A n
v
w

2
2
2
1
1
B
B
B n
v
w

2
2
2
1
1
C
C
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B
B
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Geodesic
• Lets consider the distance measured from P1 to P2
and P2 to P1
• Plane containing P2 and the normal at P2 cuts the
normal section P2 P1 as P2 αP1
• For reciprocal observation Plane containing P1
and the normal at P1 cuts the normal section P1
P2 as P1 γP2
α
γ
β

• The above normal sections are not coinciding
or the distance from P2 αP1 and P1 γP2 are not
the same
• Two distance are obtained in measuring from
P2 to P1 and P1 to P2
• Hence unique distance between two points is
defined on the spheroid
• The shortest line between P1 and P2 on the
spheroid is called the geodesic.
• That is the line from P1 to P2 via β

Spherical Excess
• In triangulation of a small area with sides of
the triangle less than 3.5Km, the triangles are
considered to be plane
• When sides are more than that the sides are
considered as arcs due to the curvature of the
earth
• Sum of three angles are more than 1800 by an
amount which is known as spherical excess
• It’s value can be calculated from the following
formula

• Spherical excess in degrees =
• Spherical excess in Seconds =
Where A = area of the spherical triangle (Km2)
and R is the radius of the earth(Km)
0
2
180
R
A

"60601802

R
A

"206265
"648000
2
2


R
A
R
A


Adjustment of Quadrilateral
• Let eight angles of a quadrilateral are
measured independently
• Angles should be corrected for spherical
excess
1
8
2
3 4
5
67
• While looking from intersection points of
diagonals angles 1, 3, 5 and 7 are called right
hand angles and 2, 4, 6 and 8 are left hand angles

• Least square adjustment is selected for the
adjustment of angles
• Following conditions should be fulfilled by the
adjusted angles
Angle 1+2+3+4+5+6+7+8 = 3600
Angle 1+2 = Angle 5+6
Angle 3+4 = Angle 7+8
• If one side and the coordinates of one point are
known then other sides and coordinates can be
calculated

Base Line measurements
• Field work for base line measurement is
carried out by two groups as setting out group
and the measuring group
• After the measurements the most probable
length is calculated applying the following
corrections

Corrections for base line measurement
• Correction for the absolute length
• Correction for temperature
• Correction for pull or tension
• Correction for Sag
• Correction for slope
• Correction for alignment
• Reduction to mean sea level
• Axis signal correction
• Correction for the unequal height
• Reduction to Chord to arc

Triangulation

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