DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
CHAPTER TWO
2.1 Introduction
2 . 2-D Control Surveying
A horizontal control survey is the surveying process required to establish the horizontal positions of a
number of points located on or near the surface of the earth. Such points after their position have
been determined are referred to as horizontal control points. It is usually one of the first steps in an
engineering projects. Horizontal control points provide important link between the physical surface of
the earth and engineering design. After completion of the project, the are used to monitor the
performance of engineering facilities.horizontal control survey usually involves the measurement of
angles and distance. The horizontal positions of the points can be obtained in a number of different
ways in addition to traversing. Theses methods are triangulation, trilateration, intersection, resection
and satellite positioning.
2.2 TRIANGULATION
The method of surveying called triangulation is based on the trigonometric proposition that if one
side and two angles of a triangle are known, the remaining sides can be computed. Furthermore, if
the direction of one side is known, the directions of the remaining sides can be determined.
A triangulation system consists of a series of joined or overlapping triangles in which an occasional
side is measured and remaining sides are calculated from angles measured at the vertices of the
triangles. The vertices of the triangles are known as triangulation stations. The side of the triangle
whose length is predetermined, is called the base line. The lines of triangulation system form a
network that ties together all the triangulation stations (Fig. 1.1)
Fig. 1.1 Triangulation network
2.3.1 PRINCIPLE OF TRIANGULATION
Fig. 1.2 shows two interconnected triangles ABC and BCD. All the angles in both the triangles and the
length L of the side AB, have been measured.
Also the azimuth θ of AB has been measured at the triangulation station A, whose coordinates (XA,
YA), are known The objective is to determine the coordinates of the triangulation stations B, C, and D
by the method of triangulation.
Fig. 1.2 Principle of triangulation
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
By using the above figure (Fig 1.2) Let us first calculate the lengths of all the lines. By sine rule in
∆ABC, we have
AB
BC
CA


Sine3 Sine1 Sine2
We have
AB = L = lAB or BC 
LSin1
 l BC
Sin3
and
CA 
LSin2
 lCA
Sin3
Now the side BC being known in ∆BCD, by sine rule, we have
BC
CD
BD


Sine6 Sin4 Sin5
We have BC 
LSin1
 LSin1  Sin4
 lCD

 l BC or CD  
 Sin3  Sin6
Sin 3
and
 LSine1  Sine5
BC  
 l BD

 Sine3  Sine6
Let us now calculate the azimuths of all the lines.
Azimuth of AB     AB
Azimuth of AC     1   AC
Azimuth of BC    180  2   BC
Azimu h of BD    180    2   4    BD
t
Azimuth of CD     2  5   CD
From the known lengths of the sides and the azimuths, the consecutive coordinates can be computed
as below.
Latitude of AB  l AB Cos AB  l AB
Departure of AB  l AB Sin AB  D AB
Latitude of AC  l AC Cos AC  LAC
Departure of AC  l AC Sin AC  DAC
Latitude of BD  l BD Cos BD  LBD
Departure of BD  l BD Sin BD  LBD
Latitude of CD  lCDCos CD  LCD
Departure of CD  lCD Sin CD  DCD
The desired coordinates of the triangulation stations B, C, and D are as follows
X-coordinate of B, X B  X A  D AB
Y-coordinate of B, YB  YB  L AB
X-coordinate of C, X C  X A  DAC
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
Y-coordinate of C, YC  YA  LAC
X-coordinate of D, X D  X B  DBD
Y-coordinate of D, YD  YB  LBD
It would be found that the length of side can be computed more than once following different routes,
and therefore, to achieve a better accuracy, the mean of the computed lengths of a side is to be
considered.
2.3.2 OBJECTIVES OF TRIANGULATION SURVEYS
The triangulation surveys are carried out
 to establish accurate control for plane and geodetic surveys of large areas, by terrestrial
methods,
 to establish accurate control for photogrammetric surveys of large areas,
 to assist in the determination of the size and shape of the earth by making observations for
 latitude, longitude and gravity, and
 to determine accurate locations of points in engineering works such as :
Fixing centre line and abutments of long bridges over large rivers,
Fixing centre line, terminal points, and shafts for long tunnels,
Transferring the control points across wide sea channels, large water bodies, etc.,
Detection of crustal movements, etc.
Finding the direction of the movement of clouds.
2.3.3 CLASSIFICATION OF TRIANGULATION SYSTEM
Based on the extent and purpose of the survey, and consequently on the degree of accuracy desired,
triangulation surveys are classified as first-order or primary, second-order or secondary, and
third-order or tertiary. First-order triangulation is used to determine the shape and size of the earth
or to cover a vast area like a whole country with control points to which a second-order triangulation
system can be connected. A second-order triangulation system consists of a network within a
first-order triangulation. It is used to cover areas of the order of a region, small country, or province. A
third-order triangulation is a framework fixed within and connected to a second-order triangulation
system. It serves the purpose of furnishing the immediate control for detailed engineering and
location surveys.
2.3.4 TRIANGULATION FIGURES AND LAYOUTS
The basic figures used in triangulation networks are the triangle, braced or geodetic quadilateral, and
the polygon with a central station
Fig. 1.3 Basic triangulation figures
The triangles in a triangulation system can be arranged in a number of ways. Some of the commonly
used arrangements, also called layouts, are as follows :
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
1. Single chain of triangles
When the control points are required to be established in a narrow strip of terrain such as a valley
between ridges, a layout consisting of single chain of triangles is generally used as shown in Fig. 1.4.
This system is rapid and economical due to its simplicity of sighting only four other stations, and does
not involve observations of long diagonals. On the other hand, simple triangles of a triangulation
system provide only one route through which distances can be computed, and hence, this system
does not provide any check on the accuracy of observations. Check base lines and astronomical
observations for azimuths have to be provided at frequent intervals to avoid excessive accumulation
of errors in this layout.
Fig. 1.4 Single of triangles
2. Double chain of triangles
A layout of double chain of triangles is shown in Fig. 1.5. This arrangement is used for covering the
larger width of a belt. This system also has disadvantages of single chain of triangles system.
3. Braced quadrilaterals
Fig. 1.5 Double chain of triangles
A triangulation system consisting of figures containing four corner stations and observed diagonals
shown in Fig. 1.6, is known as a layout of braced quadrilaterals. In fact, braced quadrilateral consists
of overlapping triangles. This system is treated to be the strongest and the best arrangement of
triangles, and it provides a means of computing the lengths of the sides using different combinations
of sides and angles. Most of the triangulation systems use this arrangement.
Fig. 1.6 Braced quadrilaterals
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
4. Centered triangles and polygons
A triangulation system which consists of figures containing interior stations in triangle and polygon as
shown in Fig. 1.7, is known as centered triangles and polygons.
Fig. 1.7 Centered triangles and polygons
This layout in a triangulation system is generally used when vast area in all directions is required to be
covered. The centered figures generally are quadrilaterals, pentagons, or hexagons with central
stations. Though this system provides checks on the accuracy of the work, generally it is not as strong
as the braced quadrilateral arrangement. Moreover, the progress of work is quite slow due to the fact
that more settings of the instrument are required.
5. A combination of above system
Sometimes a combination of the above systems may be used wich may be according to the shape of
the area and the accuracy requirements.
2.3.5 CRITERIA FOR SELECTION OF THE LAYOUT OF TRIANGLES
The under mentioned points should be considered while deciding and selecting a suitable layout of
triangles
1. Simple triangles should be preferably equilateral.
2. Braced quadrilaterals should be preferably approximate squares.
3. Centered polygons should be regular.
4. The arrangement should be such that the computations can be done through two or more
independent routes.
5. The arrangement should be such that at least one route and preferably two routes form wellconditioned triangles.
6. No angle of the figure, opposite a known side should be small, whichever end of the series is used
for computation.
7. Angles of simple triangles should not be less than 45°, and in the case of quadrilaterals, no angle
should be less than 30°. In the case of centered polygons, no angle should be less than 40°.
8. The sides of the figures should be of comparable lengths. Very long lines and very short lines
should be avoided.
9. The layout should be such that it requires least work to achieve maximum progress.
10. As far as possible, complex figures should not involve more than 12 conditions.
It may be noted that if a very small angle of a triangle does not fall opposite the known side it does
not affect the accuracy of triangulation.
2.3.6 WELL-CONDITIONED TRIANGLES
The accuracy of a triangulation system is greatly affected by the arrangement of triangles in the
layout and the magnitude of the angles in individual triangles. The triangles of such a shape, in which
any error in angular measurement has a minimum effect upon the computed lengths, is known as
well-conditioned triangle.
In any triangle of a triangulation system, the length of one side is generally obtained from
computation of the adjacent triangle. The error in the other two sides if any, will affect the sides of
the triangles whose computation is based upon their values. Due to accumulated errors, entire
triangulation system is thus affected thereafter. To ensure that two sides of any triangle are equally
affected, these should, therefore, be equal in length. This condition suggests that all the triangles
must, therefore, be isoceles.
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
Let us consider an isosceles triangle ABC whose one side AB is of known length (Fig. 1.10). Let A, B,
and C be the three angles of the triangle and a, b, and c are the three sides opposite to the angles,
respectively.
Fig. 1.10 Triangle in a triangulation system
As the triangle is isosceles, let the sides a and b be equal. Applying sine rule to ∆ABC, we have
a
c

SinA SinC
……………………………………………..1
OR
ac
SinA ……………………………………………………..2
SinC
If an error of δA in the angle A, and δC in angle C introduce the errors
respectively, in the side a, then differentiating Eq. (2) partially, we get
 a1  c
δa1
and
CosA A
…………………………………………………3
SinC
and
 a 2  c
SinACosC c ………..………………………………..4
Sin 2 C
Dividing Eq. (3) by Eq. (2), we get
 a1
  ACotA ………………..…………………………………………5
a
Dividing Eq. (4) by Eq. (2), we get
 a2
  C CotC
a
…………..…………………………………………….6
If δA = δC = ±α, is the probable error in the angles, then the probable errors in the side a are
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δa2 ,
DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
a
  Cot 2 A  Cot 2 C
a
But
Or
C  180  ( A  B)
 180   2 A
A being equal to B.
a
  Cot 2 A  Cot 2 2 A ………………………………..………7
a
 a to be minimum,
Cot 2 A  Cot 2 2 A should be a minimum.
a
Differentiating Cot 2 A  Cot 2 2 A with respect to A, and equating to zero, we have
From( Eq. .7), we find that, if
4 Cos 4 A  2 Cos 2 A  1  0 ……………………………..8
Solving Eq. (8), for Cos A, we get
A = 56°14' (approximately)
Hence, the best shape of an isoceles triangle is that triangle whose base angles are 56°14' each.
However, from practical considerations, an equilateral triangle may be treated as a well-conditional
triangle. In actual practice, the triangles having an angle less than 30° or more than 120° should not
be considered.
2.3.7 STRENGTH OF FIGURE
The strength of figure is a factor to be considered in establishing a triangulation system to maintain
the computations within a desired degree of precision. It plays also an important role in deciding the
layout of a triangulation system.
It is based on the fact that computations in triangulation involve use of angles of triangle and length
of one known side. The other two sides are computed by sine law. For a given change in the angles,
the sine of small angles change more rapidly than those of large angles. This suggests that smaller
angles less than 30° should not be used in the computation of triangulation. If, due to unavoidable
circumstances, angles less than 30° are used, then it must be ensured that this is not opposite the
side whose length is required to be computed for carrying forward the triangulation series.
The expression for evaluation of the strength of figure is for the square of the probable error (L²)
that would occur in the sixth place of the logarithm of any side, if the computations are carried from a
known side through a single chain of triangles after the net has been adjusted for the side and angle
conditions. The expression for L² is
L2 
4 2
d R ……………………………………………..9
3
where d is the probable error of an observed direction in seconds of arc, and R is a term which
represents the shape of figure. It is given by
R
DC
( A2   A B   B2 ) …………………………………10

D
where
D = the number of directions observed excluding the known side of the figure,
 A ,  B ,  C = the difference per second in the sixth place of logarithm of the sine of the distance
angles A, B and C, respectively. (Distance angle is the angle in a triangle opposite to aside), and
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DDIT Surveying Engineering Department
Introduction to Geodesy(SEng 3031)
C = the number of geometric conditions for side and angle to be satisfied in each figure. It is given
By
C = (n' – S' + 1) + (n – 2S + 3)………………………………..11
where
n = the total number of lines including the known side in a figure,
n' = the number of lines observed in both directions including the known side,
S = the total number of stations, and
S' = the number of stations occupied.
In any triangulation system more than one routes are possible for various stations. The strength of
figure decided by the factor R alone determines the most appropriate route to adopt the best shaped
triangulation net route. If the computed value of R is less, the strength of figure is more and vice
versa.
2.3.8 ROUTINE OF TRIANGULATION SURVEY
The routine of triangulation survey, broadly consists of
(a) field work, and (b) computations.
The field work of triangulation is divided into the following operations :
(i) Reconnaissance
(ii) Erection of signals and towers
(iii) Measurement of base line
(iv) Measurement of horizontal angles
(v) Measurement of vertical angles
(vi) Astronomical observations to determine the azimuth of the lines.
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