- 2. Triangulation – An Introduction Triangulation is one of the methods of fixing accurate controls. It is process of measuring the angles of a chain or a network or triangles formed by stations marked on the surface of the earth. where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles, while R is the radius of the triangle's circumcircle. A triangulation system consists of a series of joined or overlapping triangles in which an occasional side called as base line, is measured and remaining sides are calculated from the angles measured at the vertices of the triangles, vertices being the control points are called as triangulation stations. It is based on the trigonometric proposition that if one side and two angles of a triangle are known, the remaining sides can be computed. The method of surveying by triangulation was first introduced by the Dutchmen Snell in 1615.
- 3. Triangulation – An Introduction The side of the first triangle, whose length is predetermined is called the base line and vertices of the individual triangles are known as triangulation stations. To minimize accumulation of errors in lengths, subsidiary bases at suitable intervals are provided and to control error in azimuth of stations, astronomical observation are made at intermediate stations. The triangulation stations at which astronomical observations for azimuth are made, are called Laplace stations.
- 4. Triangulation – Purpose and Objectives The purpose of triangulation surveys The main objective of triangulation 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. 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. 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.
- 5. Triangulation – Classification • Primary Triangulation or First order Triangulation. • Secondary Triangulation or Second order Triangulation. • Tertiary Triangulation or Third order Triangulation. Secondary Triangulation It is triangulation system which is employed to connect two primary series and thus to provide control points closer together than those of primary triangulation. If any triangulation series which is carried out as primary does not attain the standard of accuracy of that class, due to unfavorable conditions, may also be classified as triangulation of second order. Tertiary Triangulation It is the triangulation system which is employed to provide control points between stations of primary and second order series. In the department of Survey of India, tertiary triangulations, known as topo triangulation, form the immediate control for topographical surveys on various scales. On the basis of quality, accuracy and purpose, triangulations are classified as : Primary Triangulation It is the highest grade of triangulation system which is employed either for the determination of the shape and size of the earth’s surface or for providing precise planimetric control points on which subsidiary triangulations are connected. The stations of first order triangulation are generally selected 16 to 150 km apart. Every possible precaution is taken in making linear, angular and astronomical observation, and also in their computation.
- 7. Triangulation – Laying out The arrangement of the triangles of a series is known as the layout of triangulation. A series of triangulation may consists of either of the following orders shown. This layout of triangulation is generally used when control points are provided in a narrow strip of terrain such as a valley between ridges. 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 do not provide any check on the accuracy of observations as there is only one route through which distances can be computed. To avoid excessive accumulated errors, check base lines and astronomical observations for azimuth at frequent intervals are therefore very necessary, in this layout. A triangulation system which consists of figures containing four corner stations and observed diagonals is known as a layout of braced quadrilaterals. This system is treated to be the best arrangement of triangles as it provides a means of computing the lengths of sides using different combination of sides and angles. Simple chain of triangles Braced quadrilateral A triangulation system which consists of figures containing centered polygons and centered triangles is known as centered triangles and polygons. This layout of triangulation 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. Centred triangles and polygons
- 8. Triangulation – Laying out The arrangement of the triangles of a series is known as the layout of triangulation. A series of triangulation may consists of either of the following orders shown. 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 center of the country. 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. 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. 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. Grid Iron System Central System
- 9. Triangulation – Great Trigonometrical Survey Source: Raman, Anantanarayanan, and Vancheeswar Balakrishnan. "The spark that fired the Great Trigonometrical Survey of India." Current Science 118.1 (2020): 147-154.
- 10. Triangulation – Ideal figures Factors to be considered while deciding and selecting a particular figure in any triangulation system. • Simple triangles should be preferably equilateral. • Braced quadrilaterals should be preferably square. • Centred polygons should be regular. • Angles of simple triangles should not be less than 45° and in case of quadrilaterals no angle should be less than 30°. • In case of centred polygons, no angle should be less than 40°. • The sides of the figure should be of comparable length. Well-conditioned triangle 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 isosceles. Hence, the best shape of an isosceles 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.
- 11. Triangulation – Field Procedures & Measurements To carry out fields work of triangulation, following steps are involved • Reconnaissance. • Erection of signals. • Measurement of the base lines. • Measurement of horizontal angles. Reconnaissance Preliminary field inspection of the entire area to be covered by triangulation is known as ‘reconnaissance’. During reconnaissance, the surveyor goes over the area and decides the best plan of working, keeping in view the main principle of surveying, i.e., working, from the whole to the part. The reconnaissance survey, thus requires great experience, judgement and skill. The accuracy and economy of triangulation depends upon the points reconnaissance survey. In includes the following operations : • Proper examination of the terrain. • Selection of suitable positions for base lines. • Selection of suitable positions of triangulation stations. • Determination of intervisibility of triangulation stations. • Selection of conspicuous well defined natural points to be used • as intersected points, the points observed from two or three stations. Triangulation stations are selected, keeping in view the following considerations. • Intervisibility of triangulation stations. For this purpose, stations are placed on the highest point of elevated places such as hill tops, house tops, etc. • Easy access to the stations with instruments. Various triangulation stations should form we l conditioned triangles. • Stations should be useful for providing intersected points and also for details survey. • For plane surveys, excessively distant stations should be avoided. • Stations should be on commanding situations so that these may be used for further extension of the triangulation system. • Grazing rays (line of sights) should be avoided and no line of sight should pass over the industrial areas to avoid irregular atmospheric refraction. Selection of Triangulation Stations During reconnaissance the exact positions of various triangulation stations are permanently marked on the ground so that the theodolite and signal may be centered accurately over them. Marking of stations
- 12. Triangulation – Field Procedures & Measurements Types of signals Erection of signals To define exact position of triangulation station during observations from other stations, signals are used. Various types of signals are centred vertically over the station marks and observations are made to these signals. It is very necessary to ensure that signals whenever used are truly vertical centred over the station marks. The accuracy of triangulation is entirely dependent on the degree of accuracy of centring the signals. Greatest care of centring the transit over the station mark will be useless, unless some degree of care of centering the signals is impressed upon. Requirements of an Ideal Signal A good signal should fulfil the following requirements: • It should be conspicuous i.e., it should be clearly visible from a distance against any background. • It should provide easy and accurate bisection by telescope. • It should be capable of being accurately centred over the station mark. • It should exhibit very little phase error of bisection of the signal.