2. Introduction
• The horizontal positions of points is a network developed to provide
accurate control for topographic mapping, charting lakes, rivers and ocean
coast lines, and for the surveys required for the design and construction of
public and private works of large extent. The horizontal positions of the
points can be obtained in a number of different ways and are:
1. Traversing
2. Triangulation
3. Trilateration
4. Intersection
5. Resection and
6. Satellite positioning
3. Triangulation
• a method of surveying 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.
4. Triangulation
• 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).
6. Trilateration
• A trilateration system also consists of a series of joined or
overlapping triangles. However, for trilateration the lengths
of all the sides of the triangle are measured and few
directions or angles are measured to establish azimuth.
Trilateration has become feasible with the development of
electronic distance measuring (EDM) equipment which has
made possible the measurement of all lengths with high
order of accuracy under almost all field conditions.
7. Trilateration
• A combined triangulation and trilateration system consists of a
network of triangles in which all the angles and all the lengths are
measured.
• Such a combined system represents the strongest network for
creating horizontal control
• Since a triangulation or trilateration system covers very large area, the
curvature of the earth has to be taken into account. These surveys are,
therefore, invariably geodetic.
• Triangulation surveys were first carried out by Snell, a Dutchman, in
1615. Field procedures for the establishment of trilateration station are
similar to the procedures used for triangulation, and therefore,
henceforth in this chapter the term triangulation will only be used.
8. 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. Let us first calculate the lengths of
all the lines.
13. PRINCIPLE OF TRIANGULATION
• From the known length of the sides and azimuths, the
consecutive and independent coordinates can be computed
as:
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.
14. OBJECTIVES 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.
15. OBJECTIVES OF TRIANGULATION
SURVEYS
• The triangulation surveys are carried out
• (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
16. OBJECTIVES OF TRIANGULATION
SURVEYS
(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.
17. 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.
19. CLASSIFICATION OF TRIANGULATION
SYSTEM
• 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
21. TRIANGULATION FIGURES AND
LAYOUTS
• 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. Central triangles and polygons
5. A combination of above systems.
22. 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
24. Double chain of triangles
• A layout of double chain of triangle is shown in fig 1.5. This
arrangement is used for covering the larger width of belt.
This system also has disadvantages of single chain of
triangles system
25. Braced Quadrilaterals
• 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.
27. 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.
• 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.
29. A 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
30. LAYOUT OF PRIMARY TRIANGULATION
FOR LARGE COUNTRIES
• The following two types of frameworks of primary
triangulation are provided for a large country to cover the
entire area.
1. Grid iron system
2. Central system
31. Grid Iron System
• In this system, the primary triangulation is laid in series of chains
of triangles, which usually runs roughly along meridians (north-
south) and along perpendiculars to the meridians (east-west),
throughout the country (Fig. 1.8).
• The distance between two such chains may vary from 150 to 250
km.
• The area between the parallel and perpendicular series of
primary triangulation, are filled by the secondary and tertiary
triangulation systems.
• Grid iron system has been adopted in India and other countries
like Austria, Spain, France, etc.
33. Central system
• In this system, the whole area is covered by a network of
primary triangulation extending in all directions from the
initial triangulation figure ABC, which is generally laid at the
centre of the country (Fig. 1.9).
• This system is generally used for the survey of an area of
moderate extent. It has been adopted in United Kingdom
and various other countries.
35. 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 well- conditioned triangles.
36. CRITERIA FOR SELECTION OF THE
LAYOUT OF 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.
37. 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. 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.
42. WELL-CONDITIONED TRIANGLES
• Hence, the best shape of an isosceles triangle is that
triangle whose base angels are 56⁰14’ 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 30⁰ or more than 120⁰
should not be considered.
