Triangulation is the process of tracing and measurement

Chapter 2
TRIANGULATION AND TRILATERATION
2.1 Introduction
2.2 Accuracy of horizontal control systems
2.3 Triangulation figures; Choice of figures; Triangulation
procedure
2.4 Angle and side conditions; error propagation in
triangulation; strength of figure in triangulation; base line
2.5 Triangulation computations; Adjustment of triangulation
2.6 Trilateration; adjustment of trilateration
2.7 Combined triangulation and trilateration
2.8 First and second geodetic problem on the plane, trig
sections, station and target eccentricities
2.9 Field works

Chapter 2
Triangulation:
Triangulation is the process of
measuring the angle of the
network of the triangles formed
by the station on the surface of
the earth.
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. 8.1).
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Base Line
Triangulation
Station
Fig 8.1 Triangulation Network
E
D
A
F
B
C

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Chapter 2

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We have from fig.
n = CA sinβ … … … (i)
Also from sine law,
Re-arrange to obtain,
From Eqn. (i)
Base Line
Triangulation
Station
Fig 8.1 Triangulation Network
E
D
A
F
B
C
A
B
C
d
n
α
β
π-(α+β)
AB
BC
CA
)}
(
sin{
sin
sin 



 



)}
(
sin{
sin






d
CA
)}
(
sin{
sin
sin








d
n
β)
sin(α
sinβ
sinα
d
n




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Trilateration:
Trilateration is the process of
determining absolute or relative
locations of points by measurement
of distances.
In trilateration no angular measurement
is needed but angle is computed by
using cosine rule. With the advent of
Electronic Distance Measuring (EDM)
equipment, trilateration has wide
application and can totally replace
triangulation.
B
A C
Triangle
a
b
c
Similarly, CosB and CosC can be calculated.
𝑪𝒐𝒔𝑨=
𝒃𝟐
+𝒄𝟐
−𝒂𝟐
𝟐𝒃𝒄

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Principles and classifications of Triangulation system
The principle of the triangulation is based upon the properties
of triangle. If the three angles and the length of one side of a
triangle are known, then the lengths of the remaining sides of
the triangle can be computed by trigonometry.
The side of first triangle, whose length is measured, is called
the base line
By sine rule in ΔABC, we have

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Now the side BC being known in ΔBCD, by sine rule, we have































sin6
sin5
sin3
Lsin1
BD
and,
sin6
sin4
sin3
Lsin1
CD
Now,
sin3
Lsin1
BC
Wehave,
sin5
BD
sin4
CD
sin6
BC

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Example: In a given trilateration network as shown in figure has the
following distance observation AB =7954.20m, BC = 4464.12m, AC =
11405.77m. For given coordinates of A and B, determine the co-
ordinates of station C. [2071 back)
C
B
A
Station Easting (m) Northing (m)
A 351240.22 3038628.80
B 356788.67 3044328.27

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Example: For a given trilateration, station A, B and C are three
visible stations from O. The computed sides of the triangle ABC are
AB = 1200m, BC = 1400m and CA = 1950m. A station O is stablished
outside the triangle and its position is to be determined by
resection on A, B and C, the angle AOB and BOC being respectively.
430
30’ and 450
10’. Determine distances of OA and OC
C
B
A
O

Objective of Triangulation Surveys
The main objective of triangulation or trilateration surveys is to provide a number of
stations whose relative and absolute positions, horizontal as well as vertical, are
accurately established. More detailed location or engineering survey is then carried
out from these stations.
The triangulation surveys are carried out (purpose of triangulation)
(i) To establish accurate control for plane and geodetic surveys of large areas,
by terrestrial methods,
(ii) To establish accurate control for photogrammetric surveys of large areas,
(iii) To assist in the determination of the size and shape of the earth by making
observations for latitude, longitude and gravity, and
(iv) To determine accurate locations of points in engineering works such as :
(a) Fixing centre line and abutments of long bridges over large rivers.
(b) Fixing centre line, terminal points, and shafts for long tunnels.
(c) Transferring the control points across wide sea channels, large water
bodies, etc.
(d) Detection of crustal movements, etc.
(e) Finding the direction of the movement of clouds.
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Classification of Triangulation System
On the basis of quality, accuracy, and purpose, triangulations are classified
as:
a) Primary Triangulation or First Order Triangulation
b) Secondary Triangulation or Second Order Triangulation
c) Tertiary Triangulation or Third Order Triangulation
a) Primary Triangulation or First Order Triangulation
First-order triangulation is the highest grade of triangulation system. This
system of triangulation is provided to determine the shape and size of
the earth’s surface or to provide the precise control points on which
secondary and tertiary triangulations are connected. The primary
triangulation stations are selected 16 to 150 Km apart. Every possible
precaution is taken in making linear, angular and astronomical
observations, and also in their computation.

b) Secondary Triangulation Or Second Order Triangulation
It is the triangulation system which is employed to connect two primary
series and thus to provide control points closer together than those
of primary triangulation. Sometime the accuracy and quality of the
triangles cannot be achieved for primary triangulation due to the obstacles.
In such case these triangulation system may be classified as secondary
triangulation system.
It is used to cover areas of the order of a region, small country, or province.
c) Tertiary Triangulation or Third Order Triangulation
A third-order triangulation is the triangulation system, which is employed to
provide control points between stations of primary and second order
series. It serves the purpose of furnishing the immediate control for
detailed engineering and location surveys.
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Properties of triangulation systems

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Triangulation Figures and Layouts
The basic figures used in triangulation networks are the triangle, braced or
geodetic quadrilateral, and the polygon with a central station (Fig.).
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:
1. Single chain of triangles
2. Double chain of triangles
3. Braced quadrilaterals
4. Centered triangles and polygons
5. A combination of above systems.

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1) Simple (single) chain triangles
It is generally used when the control
points are provided in a narrow strip of
terrain such as a valley between ridges.
This system is rapid and economical than
other systems. Simple triangles do not
provide any check on the accuracy of
observations as there is only one route
through which distances can be
A
B
C
D
E
F
G

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A B
D
C
F
I
G H
E
2) Double chain of triangles
A layout of double chain of triangles is shown
in Fig. 8.5. This arrangement is used for
covering the larger width of a belt. This
system also has disadvantages of single
chain of triangles system.
A
D
E
B
C
F
3) Braced quadrilaterals

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4) Centered triangles and polygons
It consists of figures containing centered polygons and centered triangles.
This layout is generally used when vast area in all dimensions is required
to be covered. The centered figures generally are quadrilaterals,
pentagons, or hexagons with central stations. Though this system provides
proper check on the accuracy of the work, the progress of the work is
generally low due to the fact that more settings of the instrument are
required. Centered polygons should be regular as far as possible.
A
D
B
C
F
G
H
I J
K
5) Combination of all above systems
Sometimes a combination of above systems may be used which may be
according to the shape of the area and the accuracy requirements

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Criteria for selection of the arrangement 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 well conditioned triangles.
6. 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°.
7. The sides of the figures should be of comparable lengths. Very
long lines and very short lines should be avoided.

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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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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. As the triangle is isosceles,
let the sides a and b be equal. Applying
sine rule to ∆ABC, we have

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Differentiating cot²A + cos² 2A with respect to A, and equating to zero,
we have
4 cos4A + 2 cos²A – 1 = 0 ...(1.8)
Solving Eq. (1.8), for cos A, we get
A = 56°14' (approximately)

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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.

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STRENGTH OF FIGURE

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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.
The U.S. Coast and Geodetic Surveys has developed a convenient method of
evaluating the strength of a triangulation figure. 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 given by the U.S. Coast and Geodetic Surveys 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:

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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:
Where,
D = the number of directions observed excluding the known side of the
figure,
δA,δB,δC = 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 a side), and 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)
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.

Triangulation is the process of tracing and measurement

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