The triangulation Survey and trilateration Survey, are described below one by one in detail.

## Triangulation

To prevent accumulation of errors, it is necessary to provide a number of control points all over the area, which will form a framework on which the entire survey is to be based and the provision of control points can be made either by one or a combination of both the following methods,

**Theodolite****Triangulation**

The process of measuring the angle of a chain or interconnected network of the triangles formed by a number of triangulation stations marked on the surface of the earth is called triangulation.

Triangulation is more accurate than the theodolite transverse, as there is less accumulation of error than that in theodolite traverse.

The method of surveying by triangulation was first introduced by the dutchmen Snell in 1615.

## Trilateration

The type of triangulation in which the sides of the triangle instead of the angles are measured is called trilateration. In trilateration, the length of the side of the triangle is determined with the help of electronic distance measuring (EDM) devices.

The three angles of the triangles are computed trigonometrically from their observed sides.

Knowing the measured sides, a starting azimuth, and known coordinates of a starting station, the coordinates of the remaining stations can be computed.

## Measurement of the sides

**1) Measure distance i.e sloping distance and following correction is applied**

- Instrument constant
- Atmospheric correction
- Slope correction
- Chord to arc correction
- MSL correction

**2) Calculation of angles of triangles**

**3) Calculation of coordinates**

Knowing the length and bearing of each side of the triangles and known coordinates of a starting station, the latitude and departure may be calculated.

## Principle of triangulation

If all three angles and the length of one side of the triangle are known, the length of the remaining sides of the triangle can be calculated trigonometrically.

Again if the coordinates of any vertex of the triangles and azimuth of any side are also known, then the coordinates of the remaining vertices may be computed.

**Baseline:-** The side of the first triangle, whose length is pre-determined is called the baseline.

**Triangulation station: **The vertices of the individual triangles are known stations. To minimize accumulation error in length, subsidiary bases at suitable intervals are provided and to control the error in the azimuth of the station, astronomical observations are made at intermediate stations.

**Laplace station: **The triangulation station at which astronomical observation for azimuth is made is called Laplace station.

## Purpose of triangulation:

Triangulation surveys are carried out for:

1. Establishment of accurate control points for plane and geodetic surveys of large areas by the ground method.

2. Establishment of accurate control points for a photogrammetric survey of large areas.

3. Locating engineering works accurately i.e.

a. Fixing the centerline, terminal points, and shaft for long tunnels.

b. Fixing the centerline, piers, and abutments of the long bridges over rivers.

c. Transferring the control points across wide sea channels, large water bodies, etc.

d. Finding the direction of the movement of clouds.

## Classification of triangulation:

On the basis of quality, accuracy, and purpose, triangulations are classified as:

1. Primary triangulation or 1″ order triangulation

2. Secondary triangulation or 2nd order triangulation

3. Tertiary triangulation or 3d order triangulation

#### 1) Primary triangulation:

It is the highest grade or order of triangulation system which is carried out either for the determination of the shape and size of the earth’s surface or for providing precise planimetric control points for the subsidiary or secondary triangulation is called primary triangulation.

The stations of the first-order triangulation are generally selected 30 km to 150 km apart,

#### General specification of primary triangulation:

1. Length of the baseline: 5 km to 15 km

2. Length of the sides of the triangle: 30 km to 150 km

3. Average triangular error or closer: Less than 1 sec

4. Maximum station closer: not more than 3 sec

5. Actual error of base: 1:300000

6. Probable error of base: 1 in 1000000

7. Discrepancy between two measures: 10mm/km

8. Probable error of computed distance: 1 in 60000 to 1 in 250000

9. Probable error in astronomical observation: 0.5 sec

#### 2) Secondary triangulation

The triangulation system which is employed to connect two primary series and thus to provide control points closer together than those of primary triangulation is called secondary triangulation

The size of the triangle formed by secondary triangulation is smaller than the primary triangulation.

#### General specification of secondary triangulation

1. Length of the baseline: 1.5 km to 5 km

2. Length of the sides of the triangle: 8 km to 65 km

3. Average triangular error or closer : 3 sec

4. Maximum station closer: 8 sec

5. Actual error of base: 1:150,000

6. Probable error of base: 1 in 5,00,000

7. Discrepancy between two measures: 20 mm/km

8. Probable error of computed distance: 1 in 20,000 to 1 in 50,000

9. Probable error in astronomical observation: 2 sec

#### 3) Tertiary triangulation

The triangulation system which is employed to provide a control point between stations of primary and secondary triangulation series is called tertiary triangulation.

#### General specification of Tertiary triangulation

1. Length of the baseline: 0.5 km to 3 km

2. Length of the sides of the triangle: 1.5 km to 10 km

3. Average triangular error or closer: 6 sec

4. Maximum station closer: 12 sec

5. Actual error of base: 1:75,000

6. Probable error of base: 1 in 250,000

7. Discrepancy between two measures: 25 mm/km

8. Probable error of computed distance: 1 in 5,000 to 1 in 20,000

9. Probable error in astronomical observation: 5 sec

## Layout of triangulation

The arrangement of the triangles of a series is known as the layout of triangulation. A series of triangulation may consist of:

- Simple triangles in chain
- Braced quadrilaterals
- Centered triangles and polygons

#### Simple triangles in chain

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 paid and economical due to its simplicity of sighting only four other stations and does not observation of long diagonals.

Simple triangles of a triangulation system do not provide any check on the accuracy of observations as there is only one route through which distance can be computed.

In this layout to avoid excessive accumulated errors, checking baselines and astronomical observations for an azimuth at frequent intervals are necessary.

#### Braced quadrilaterals

A triangulation system that consists of figures containing four corner stations and observed diagonals is known as a layout of braced quadrilaterals.

This system is treated to the best arrangement of triangles as it provides a means of computing the length of sides using different combinations of sides and angles.

#### Centered tiangles and polygons

A triangulation system that consists of figures containing centered polygons and centered triangles is known as centered triangles and polygons.

This layout of triangulation is generally used when the vast area in all dimensions is required to be covered.

The centered figures are generally quadrilaterals, pentagons, or hexagons with central stations.

Though this system provides a 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 is required.

## Well conditioned triangle

The shape of the triangle in which any errors in angular measurement have a minimum effect upon the length of the computed sides is known as a well-conditioned triangle of a triangulation system.

The best shape of a triangle is isosceles with base angles equal to 56 degrees 14 minutes.

Practically, an equilateral triangle is the most suitable triangle

Generally, angle smaller than 30° and greater than 120 is avoided.

## Routine of triangulation survey:

The entire routine of triangulation survey may be broadly divided into a two-step**1. Fieldwork of triangulation2. Computation of triangulation.**

1. Fieldwork of triangulation:

The following steps are involved to carry out the fieldwork of triangulation:

**a. Reconnaissance**

i. Proper examination of the terrain

ii. Selection of suitable positions for baselines

iii. Selection of suitable positions of triangulation stations

iv. Determination of intervisibility of triangulation stations

**b. Erection of signals****c. Measurement of initial baseline.**

d. Measurement of horizontal angles

e. Measurement of vertical angles.

f. Astronomical observations to find the azimuth of the sides

g. Measurement of closing baseline

## 2. Computation of triangulation:

It includes the following:

a. Checking the means of the observed angles “

b. Checking the triangulation errors

c. Checking the total round at each station (i.e. 360 degrees)

d. Computation of the corrected length of the baseline

e. Computation of the sides of the main triangles applying sine rule

f. Computation of the latitude and departure of each side of the triangulation network.

**Types of triangulation stations:**

1. Main station

2. Subsidiary station

3. Satellite station

4. Pivot station

**1. Main Station:** The triangulation station which is used to carry forward the networks of triangulation is called the main station.

**2. Subsidiary station: **The triangulation stations which are used only to provide additional rays to intersected points are known as a subsidiary stations.

**3. Satellite station:**

The triangulation stations which are selected close to the main triangulation stations to avoid intervening obstructions are known as satellite stations or eccentric or false stations.

**4. Pivot station**

The triangulation stations at which no observations are made but the angles at that station are used for the continuity of a triangulation series are known as pivot stations.

## Accuracy of triangulation

m = root mean square error of unadjusted horizontal angle (in secs.) as obtained from the triangular error

Sigma E square is the sum of the square of all the triangular errors in the triangulation series.

n = total no. of triangles in series

## Signals and Towers

Signals are used to define the exact position of the triangulation station during observations from another station

Various types of signals are centered vertically over the station marks and observations are made to these signals.

The accuracy of triangulation is entirely dependent on the degree of accuracy of centering the signals.

#### Classification of signals

**1) Luminous signals**

a) Sun Signals

b) Night signals

- Oil lamps with reflector ( used for <80 km sight)
- Acetylene lamps (used for > 80 km sight)

**2) Opaque signals or non-luminous signal**

- Pole signal
- Target signal
- Pole and brush signal
- Stone cairn
- Beacons

## Sun signals:

- Those signals which reflect the rays of the sun towards the stations of observation are known as sun signals or heliotropes.
- It can be used in clear weather.
- Heliotropes consist of a circular plane mirror with a small hole at its center and it reflects the sun’s rays.
- It consists of crosshair on sight vane.

## Night signals

It is used in night observation

## Opaque signals

It is used during day observations.

## Pole signals

- It consists of a round pole painted black and white in alternate and is supported vertically over the station marks on a tripod.
- It is suitable for sights up to 6 km.

## Target Signal

It consists of a pole carrying two square or rectangular targets placed at right angles to each other.

## Pole and brush signals

- It consists of a straight pole about 2.5 m long with a bunch of long grass tied symmetrically around the top, making a cross.
- It is erected vertically over the station mark by heaping a pile of stones up to 1.7 m around the pole.
- It is a very useful opaque signal and must be erected over every station of observation during reconnaissance.

## Stone crains

It consists of stones built up to a height of 3 meters in a conical shape. The whitewashed opaque signal is very useful if the background is dark.

## Beacons

- It consists of red and white cloth tied around the three straight poles
- This can be easily centered over the station’s marks.
- It is useful when simultaneous observations are made at both stations.

## Indivisibility of stations

**1) When there is no obstruction due to intervening ground,** the distance of visible horizon from the station of known elevation may be calculated with the formula:

Where,

h = height of station above datum

D = distance of the visible horizon

R = mean radius of the earth

m = mean coefficient of refraction

= 0.07 for sights over land and 0.08 for sights overseas.

Substituting the value of R in km, h is given by,

**h = 0.06735 D ^{2} m**

2) Let A and B be two stations having their ground portion elevated to A_{‘} and B^{‘} respectively.

Let,

h_{1} = RL of station A

h_{2} = minimum required elevation of station B so that it is visible from A

D_{1} = distance of visible horizon from A

D_{2} = distance of visible horizon from B at an elevation h_{2}

D = known distance between A and B

Now,

and,

But, D = D_{1} + D_{2}

Therefore, D_{2 }=D – D_{1}

The line of right should be at least 3 m above the point of tangency of the earth’s surface to avoid grazing sights.

## Selection of Triangulation station

Triangulation stations are selected, keeping in view the following considerations:

1. Intervisibility of triangulation stations: stations are placed on the highest points of the elevated places such as hilltops, housetops, etc.

2. Easy access to the stations with instruments and equipment.

3. Various triangulation stations should form well condition triangles.

4. Stations should be useful for providing intersected points and also for a subsequent detailed survey.

5. Excessively distant stations should be avoided for plane surveys.

6. Stations should be in commanding positions so that these may be used for further extension of the triangulation system.

7. Grazing line of sight should be avoided and no line of sight should pass over the industrial area to avoid irregular atmospheric refraction.

I hope this article on “**Triangulation Survey and trilateration Survey**” remains helpful for you.

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