80_fsl_ch_06.pdf

53
Introduction In Chapter 3, we discussed compass surveying. A compass measures the direction by measuring the angle between the line and a reference direction, which is the magnetic meridian. A compass can measure angles up to an accuracy of 30¢ and by judgement up to an accuracy of 15¢. The principle of working of the compass is based on the property of the magnetic needle, which when freely suspended, takes the north–south direction. Compass measurements are thus affected by external chapter, we will discuss another method of measuring directions of lines; a theodolite is very commonly used to measure angles in survey work. There are a variety of theodolites—vernier, optic, electronic, etc. The improve- ments (from one form to the other) have been made to ensure ease of operation, better accuracy, and speed. Electronic theodolites display and store angles at the press of a button. This data can also be transferred to a computer for further processing. We start our discussion with the simplest theodolite—the vernier theodolite. Other forms of theodolites are discussed at the end of this chapter. The vernier theodolite is a simple and inexpensive instrument but very valuable in terms of measuring angles. The common vernier theodolite measures angles up to an accuracy of 20¢¢ CHAPTER 6 Theodolite Surveying Learning Objectives theodolite state the fundamental lines of a theodolite Preview - Copyrighted Material

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Page 1: 80_FSL_Ch_06.pdf

Introduction

In Chapter 3, we discussed compass surveying. A compass measures the direction

by measuring the angle between the line and a reference direction, which is the

magnetic meridian. A compass can measure angles up to an accuracy of 30¢ and

by judgement up to an accuracy of 15¢. The principle of working of the compass is

based on the property of the magnetic needle, which when freely suspended, takes

the north–south direction. Compass measurements are thus affected by external

chapter, we will discuss another method of measuring directions of lines; a theodolite

is very commonly used to measure angles in survey work.

There are a variety of theodolites—vernier, optic, electronic, etc. The improve-

ments (from one form to the other) have been made to ensure ease of operation,

better accuracy, and speed. Electronic theodolites display and store angles at the press

of a button. This data can also be transferred to a computer for further processing.

We start our discussion with the simplest theodolite—the vernier theodolite. Other

forms of theodolites are discussed at the end of this chapter.

The vernier theodolite is a simple and inexpensive instrument but very valuable

in terms of measuring angles. The common vernier theodolite measures angles up

to an accuracy of 20¢¢

CHAPTER

6Theodolite Surveying

Learning Objectives

theodolite state the fundamental lines of a theodolite

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Theodolite Surveying 249

in a compass, where the line of sight is simple, restricting its range, theodolites

are provided with telescopes which provide for much greater range and better ac-

curacy in sighting distant objects. It is, however, a delicate instrument and needs to

be handled carefully. The theodolite measures the horizontal angles between lines

and can also measure vertical angles. The horizontal angle measured can be the

is the angle in a vertical plane between the inclined line of sight of the instrument

and the horizontal. In the following sections we will discuss the vernier theodolite

as well as its applications in surveying.

6.1 Vernier Theodolite

The vernier theodolite is also known as a transit. In a transit theodolite or simply

transit the telescope can be rotated in a vertical plane. Earlier versions of theodolites

were of the non-transit type and are obsolete now. Only the transit theodolite will

be discussed here.

Two different views of a vernier theodolite are shown in Figs 6.1(a) and (b). The

instrument details vary with different manufacturers but the essential parts remain

the same. The main parts of a theodolite are the following.

14

12 13

119 8

10

15 166

7

5 17

2

31

18

4

18

1

32

5

76

8

10

14

11

12

15

4

1. Trivet 7. Circle plate 13. Vertical circle clamp2. Tribrach 8. A-frame 14. Telescope3. Foot screw 9. Vertical circle clamp arm 15. Plate bubble4. Inner axis 10. Vernier frame 16. Upper clamp5. Outer axis 11. Vertical circle 17. Lower clamp6. Vernier plate 12. Altitude bubble 18. Tripod head

(a) Sectional view (b) Another view

Fig. 6.1 Vernier theodolite

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250 Fundamentals of Surveying and Levelling

Levelling head The levelling head is the base of the instrument. It has the provi-

sion to attach the instrument to a tripod stand while in use and attach a plumb bob

along the vertical axis of the instrument. The levelling head essentially consists

of two triangular plates kept a distance apart by levelling screws. The upper plate

of the levelling head, also known as the tribrach, has three arms, each with a foot

screw. Instruments with four foot screws for levelling are also available. In terms

of wear and tear, the three-foot-screw instrument is preferable. The lower plate,

also known as the trivet, has a central hole and a hook to which a plumb bob can be

attached. In modern instruments, the base plate of the levelling head has two plates

which can move relative to each other. This allows a slight movement of the level-

ling head relative to the tripod. This is called a shifting head and helps in centring

the instrument over the station quickly. The functions of the levelling head are to

support the upper part of the instrument, attach the theodolite to a tripod, attach a

plumb bob, and help in levelling the instrument with the foot screws.

Lower plate The lower plate, also known as the circle plate, is an annular,

cylindrical part. The graduations are provided all around, from 0° to 360°, in the

clockwise direction. The graduations are in degrees divided into three parts so

that each division equals 20¢. An axis through the centre of the plate is known as

the outer axis or the centre. Horizontal angles are measured with this plate. The

diameter of the lower plate is sometimes used to indicate the size of or designate

the instrument; for example, a 100-mm theodolite.

Upper plate The upper plate is also a horizontal plate of a smaller diameter

attached to a solid, vertical spindle. The bevelled edge of the horizontal part

carries two verniers on diametrically opposite parts of its circumference. These

verniers are generally designated A and B. They are used to read fractions of the

horizontal circle plate graduations. The centre of the plate or the spindle is known

as the inner axis or centre. The upper and lower plates are enclosed in a metal cover

to prevent dust accumulation. The cover plate has two glass windows longer than

the vernier length for the purpose of reading. Attached to the cover plate is a metal

arm hinged to the centre carrying two magnifying glasses at its ends. The magnify-

ing glasses are used to read the graduations clearly.

Two axes or centres The inner axis as mentioned earlier is the axis of the conical

spindle attached to the upper or vernier plate. The outer axis is the centre of the

hollow cylindrical part attached to the lower or circle plate. These two axes coin-

cide and form the vertical axis of the instrument, which is one of the fundamental

lines of the theodolite.

Clamps and tangent screws There are two clamps and associated tangent or

slow-motion screws with the plates. The clamp screws facilitate the motion of the

instrument in a horizontal plane. The lower clamp screw locks or releases the lower

plate. When this screw is unlocked, the lower and upper plates move together. The

associated lower tangent screw allows small motion of the plates in the locked

position. The upper clamp screw locks or releases the upper vernier plate. When

this clamp is released (with the lower clamp locked), the lower plate does not

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Theodolite Surveying 251

move but the vernier plate moves with the instrument. This causes a change in the

reading. The upper tangent screw allows for a small motion of the vernier plate for

in the horizontal plane. The construction of the clamp and tangent screws is shown

in Fig. 6.2.

12

34

5 6 7 8 9 10 1112

13

Tangentscrew

Outer cover

Outer axis

Inner axis

Lower clampSpring

box

(a) Clamp and tangent screw

(b) Vertical circle graduations (20 main scale)¢

10

20

3040

5060

70 80

10

20

30

40

5060

70 80

0 0

Quadrants 0–90°

(c) Main circle plate graduations

90

90

Fig. 6.2 Parts of a theodolite

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252 Fundamentals of Surveying and Levelling

Plate level The plate level is a spirit level with a bubble and graduations on the

provided. The spirit level can be adjusted with the foot screw of the levelling head.

The bubble of the spirit level can be moved with the foot screws of the levelling

head, which is a very fundamental adjustment required for using the theodolite.

A small circular bubble may be provided for rough adjustment before levelling.

Index frame The index frame, also known as a T-frame or vernier frame, is a

T-shaped metal frame. The horizontal arm carries at its ends two verniers, which

C and D. The vertical leg of the T-frame, known as the clipping arm, has clipping

top of this frame. When the telescope is rotated in a vertical plane, the vertical

circle moves and vertical angles are measured on the vertical circle with the help

of these verniers.

Standard or A-frame Two standards in the shape of the letter A are attached

to the upper plate. The horizontal axis of the instrument is attached to these stan-

dards. The clipping arm of the index frame and the arm of the vertical circle clamp

are also attached to the A-frame. The A-frame supports the telescope and the

vertical circle.

Telescope The telescope is a vital part of the instrument. It enables one to see

stations that are at great distances. The essential parts of a telescope are the eye-

piece, diaphragm with cross hairs, object lens, and arrangements to focus the tele-

scope. A focusing knob is provided on the side of the telescope. Earlier, external

focusing telescopes were used. Today, only internal focusing telescopes are used

in theodolites. These reduce the length of the telescope. The telescope may carry

a spirit level on top in some instruments. The optics of the telescope and other

details have been discussed in Chapter 1.

Vertical circle The vertical circle is a circular plate supported on the trunnion

or horizontal axis of the instrument between the A-frames. The vertical circle has

a bevelled edge on which graduations are marked. The graduations are generally

quadrantal, 0°–90° in the four quadrants as shown in Fig. 6.2. The full circle system

of graduations can also be seen in some instruments. The vertical circle moves

with the telescope when it is rotated in a vertical plane. A metal cover is provided

to protect the circle and the verniers from dust. Two magnifying glasses on metal

arms are provided to read the circle and verniers. The cover has glass or plastic

Vertical circle clamp and tangent screw The vertical circle is provided with a

vertical circle, the telescope cannot be moved in a vertical plane. The tangent screw

allows for a slow, small motion of the vertical circle.

Altitude level

is used for levelling, particularly when taking vertical angle observations.

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Theodolite Surveying 253

Compass -

horizontal plates. Two plates with graduations are provided in the compass box for

ensuring that the needle ends are centred. The needle can be locked or released by

a pin. When released, the telescope can be turned in azimuth to make the north end

of the needle point to the north by making it read zero.

Tripod One accessory essential with the theodolite is the tripod on which it is

mounted when it has to be used. The tripod head is screwed onto the base or the

lower part of the levelling head. Its legs should be spread out for stability. The legs

of the tripod are also used for rough levelling.

Plumb bob A heavy plumb bob on a good string with a hook at the end is re-

hook or other device projecting from the centre of the instrument in a central

opening in the levelling head.

Main circle and vernier graduations In most of the instruments, the vernier

enables readings up to 20≤ of the arc. This is made possible by marking the gradu-

ations on the circle and the vernier suitably as follows. As shown in Fig. 6.2(b),

the main circle is graduated into degrees and each degree is divided into three

parts. Each main scale division thus represents 2¢. For the vernier, 59 main scale

divisions are taken and divided into 60 parts. 59 main scale divisions form 59 ¥20¢. Therefore, each vernier scale division represents 59 ¥ 20/60 minutes. As you

would have studied earlier, least count of the vernier = difference between a main

scale division and a vernier scale division = main scale division – vernier scale

division. Hence, in this case,

Least count = 20¢ – 59 ¥ 20/60 = 1/3 = (1/3) ¥ 60≤ = 20≤Thus the least count of the vernier in common theodolites is 20≤.

6.2 Terminology

It is important to clearly understand the terms associated with the theodolite

and its use and meaning. The following are some important terms and their

Vertical axis It is a line passing through the centre of the horizontal circle and

perpendicular to it. The vertical axis is perpendicular to the line of sight and the

trunnion axis or the horizontal axis. The instrument is rotated about this axis for

sighting different points.

Horizontal axis It is the axis about which the telescope rotates when rotated in

a vertical plane. This axis is perpendicular to the line of collimation and the verti-

cal axis.

Telescope axis It is the line joining the optical centre of the object glass to the

centre of the eyepiece.

Line of collimation It is the line joining the intersection of the cross hairs to the

optical centre of the object glass and its continuation. This is also called the line

of sight.

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254 Fundamentals of Surveying and Levelling

Axis of the bubble tube It is the line tangential to the longitudinal curve of the

bubble tube at its centre.

Centring Centring the theodolite means setting up the theodolite exactly over the

station mark. At this position the plumb bob attached to the base of the instrument

lies exactly over the station mark.

Transiting It is the process of rotating the telescope about the horizontal axis

through 180°. The telescope points in the opposite direction after transiting. This

process is also known as plunging or reversing.

Swinging It is the process of rotating the telescope about the vertical axis for

the purpose of pointing the telescope in different directions. The right swing is a

rotation in the clockwise direction and the left swing is a rotation in the counter-

clockwise direction.

Face-left or normal position This is the position in which as the sighting is done,

the vertical circle is to the left of the observer.

Face-right or inverted position This is the position in which as the sighting is

done, the vertical circle is to the right of the observer.

Changing face It is the operation of changing from face left to face right and vice

versa. This is done by transiting the telescope and swinging it through 180°.

Face-left observation It is the reading taken when the instrument is in the normal

or face-left position.

Face-right observation It is the reading taken when the instrument is in the

inverted or face-right position.

6.3 Temporary Adjustments

Theodolite has two types of adjustments—temporary and permanent. Temporary

adjustments are to be done at every station the instrument is set up. Permanent

adjustments deal with the fundamental lines and their relationships and should be

done once in a while to ensure that the instrument is properly adjusted. The funda-

mental lines and their desired relationships are explained later in this chapter and

the permanent adjustments are explained in detail in Chapter 4. In this section we

will discuss temporary adjustments.

The temporary adjustments are the following: (a) setting up and centring,

(b) levelling, (c) focusing the eyepiece, and (d) focusing the objective.

6.3.1 Setting Up and Centring The following procedure is adopted for this operation.

the station where the instrument is to be set up. The tripod legs should be well

apart and the telescope should be at a convenient height for sighting.

2. Tie a plumb bob onto the hook provided at the base. If there is no shifting head

in the instrument, centre it by adjusting the tripod legs and shifting the instru-

ment as a whole to bring the plumb bob over the station mark.

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Theodolite Surveying 255

3. To centre the plumb bob, shift the tripod legs radially as well as circumf-

erentially. Moving any leg radially shifts the plumb bob in the direction of the leg. This does not affect the level status of the instrument. Moving any leg circumferentially does not appreciably shift the plumb. However, this

movement tilts the instrument and affects the level of the plate bubbles. By

moving the legs the plumb bob is brought over the station mark at the same

time ensuring that the instrument is approximately level. This saves a lot of

time for the next operation of levelling.

(or the instrument as a whole) to bring the plumb bob over the station mark.

6.3.2 LevellingAfter setting up and centring the instrument, levelling is done. Levelling has to

be done at every station the instrument is set up. By levelling the instrument, it is

ensured that as the instrument is swung about the vertical axis, the horizontal plate

moves in a horizontal plane. The instrument may have a three-screw or a four-screw

levelling head. The levelling operations differ slightly in these two cases as detailed

in the following sections. Most instruments have only one bubble tube, but some

instruments have two bubble tubes set at right angles over the plates.

Three-screw levelling head When the theodolite has a three-screw levelling head, the following procedure is

adopted.

1. Swing the theodolite and bring the plate bubble parallel to any two of the foot

screws. Centre the bubble by rotating the foot screws. To do this, hold the foot

rotate both either inwards or outwards [see Fig. 6.3(a)]. Also note that the bubble moves in the direction

of movement of the left thumb during this operation.

2. Once the bubble traverses (or comes to the central position from the gradua-

tion of the tube), swing the instrument and bring the bubble over the third foot

screw. In this position, the bubble tube is at right angles to the earlier position.

Centre the bubble by rotating the third foot screw alone.

Foot screw

Level

Bubble

movement Level

Thumb

movement

Level

(a) (b)

Fig. 6.3 Three-foot-screw levelling head

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256 Fundamentals of Surveying and Levelling

3. Bring the plate bubble to its previous position by swinging the instrument back.

Check whether the bubble traverses. If it does not traverse, bring the bubble to

the centre using the two foot screws as before.

4. Repeat the procedure till the bubble traverses in both these positions.

5. Swing the instrument through 180° and check whether the bubble traverses.

The bubble should traverse in all positions if the instrument has been properly

adjusted.

If two plate bubbles are provided [see Fig. 6.3(b)], the procedure is the same

except that swinging the instrument through 90° is not required. When one plate

level is kept parallel to a pair of foot screws, the other plate level is over the

third foot screw (in a perpendicular direction). The third foot screw is adjusted

alternately by the same process using the foot screws over which they are paral-

lel.

Four-screw levelling head When the theodolite has a four-screw levelling head, the following procedure is

adopted.

1. After setting up and centring the theodolite, bring the plate level parallel to

any one pair of diagonally opposite foot screws. Operate these foot screws to

centre the bubble (Fig. 6.4).

2. Swing the instrument to bring the plate level parallel to the other pair of foot

screws. Centre the bubble.

3. Swing it back to the previous position. Check whether the bubble traverses.

If it does not, centre it with the foot screws to which the level is parallel.

4. Swing it back, check the position of the bubble, and repeat the procedure.

5. Once the bubble traverses in the two orthogonal positions, swing it through

180°. The bubble should traverse in this position or in any other position.

If two plate levels are provided, the procedure is the same. Bring one plate

level parallel to a pair of opposite foot screws. The other pair will be parallel to the

remaining pair of foot screws. There is no need to swing the instrument. Bring the

bubble to the central position alternately and check in the other positions.

Fig. 6.4 Four-foot-screw levelling head

6.3.3 Focusing the EyepieceFocusing the eyepiece is the operation of bringing the cross hairs to focus. The

focusing position varies with the eyesight of the observer. If the same observer is

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Theodolite Surveying 257

taking the readings, this has to be done only once. To focus the eyepiece, use the

following procedure.

1. Keep a piece of white paper in front of the telescope or direct the telescope

towards a clear portion of the sky.

2. Looking through the telescope, adjust the vision by rotating the eyepiece till

the cross hairs come into sharp and clear view.

3. If the eyepiece has graduations, note the graduation at which you get a clear

view of the cross hairs. This can help in later adjustment if required.

6.3.4 Focusing the Objective The objective lens has to be focused whenever an object is sighted, as this depends

upon the distance between the instrument and the object. A focusing screw on the

side of the telescope is operated to focus the objective. This operation brings the

image of the object in the plane of the cross hairs. This helps to exactly bisect the

object, be it a ranging rod or an arrow. To focus the objective, swing the instrument

to bring the object into view by looking over the telescope. Rotate the focusing

knob till the object is in sharp view along with the cross hairs.

6.3.5 Using the Theodolite The theodolite is mainly used to measure horizontal and vertical angles, even though

many other operations can be done with the instrument. It is a delicate and sensi-

tive instrument and needs to be handled carefully. The following points should be

noted while using the instrument.

1. The theodolite should be set up and levelled at every station. This is a funda-

mental, necessary operation and should be carried out carefully.

The line of sight is arranged to bisect the object clearly.

3. The graduated circle plate gives the outer axis and the vernier plate provides

the inner axis. Both the axes coincide if the instrument is properly adjusted

and form the vertical axis.

4. There are three clamp screws each with its own tangent screw. The lower clampscrew releases the lower plate, the upper clamp screw releases the upper ver-

nier plate, and the third vertical circular clamp releases the vertical circle. One

should be familiar with the location of the clamp screws and the corresponding

tangent screws.

5. Each clamp screw releases one plate. The lower plate is released by the lower

clamp screw. When this plate is released, swinging the instrument or rotating it

in a horizontal plane causes no change in the reading of the circle, as both the

plates move together. This is used when an object has to be sighted with the zero

setting of the circle or with any other reading without changing the reading.

6. Both the clamp screws should not be released together. When the lower clamp

is tight and the upper clamp screw is released, the upper plate moves relative to

the lower plate and the reading changes. This is done when one has to measure

an angle.

a very small movement has to be effected by the tangent screw. For each clamp

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258 Fundamentals of Surveying and Levelling

8. To set the instrument to zero at the plate circle, release the upper clamp and

rotate the instrument about the vertical axis. On the vernier A, make the zero

of the circle coincide with the zero of the vernier. Tighten the upper clamp and

using the upper tangent screw, make the zeros exactly coincide. This can be

-

tions on either side are symmetrical. Verify the condition on vernier B as well,

where the 180° graduation should coincide with the zero of the vernier.

9. While bisecting the signals or setting the zero reading, keep the line of sight

in such a position that the tangent screw moves the sight in the same direction

as the movement of the instrument. If the movement is clockwise, then the

tangent screw is adjusted to move the cross hairs from left to right.

10. Operate a tangent screw only after clamping the corresponding clamp screw.

Read the circle by bringing the glass over the reading and looking directly over

the reading to avoid any parallax error.

12. While bisecting stations with the theodolite, the station mark should be very

clear and must be a point. Bisect either the cross marks on pegs at their inter-

section or the ranging rod and arrow at their lowest pointed end.

13. Clamp screws and tangent screws need careful handling. Do not apply great

force on these screws and handle them delicately during survey work.

6.4 Measuring Horizontal Angles

To measure the horizontal angle between two lines, the following procedure is

adopted.

1. Referring to Fig. 6.5, the angle POQ is to be measured. Set up the theodolite

at O.

2. Set the instrument to read 0° 00¢ 00≤.

This is not strictly required, as the

angle can be determined as the dif-

readings. However, it is convenient

to make the initial reading zero. For

this, release the upper clamp and

rotate the instrument to make the

reading approximately zero. Clamp

the upper plate and using the upper

tangent screw, make the reading

exactly zero. Vernier A reads zero and vernier B reads 180° 00¢ 00≤.

3. Release the lower plate and rotate the instrument to bisect the station P. After

approximately bisecting it, clamp the lower plate and using the lower tangent

screw, bisect the signal exactly. The readings on the plates do not change as

both the plates move together in this operation. Check that the readings on

vernier A and B are zero and 180°, respectively.

4. Release the upper plate by loosening the upper clamp. Rotate the instrument to

screw, exactly bisect the signal at Q.

P

O

Q

Fig. 6.5 Measuring a horizontal angle

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Theodolite Surveying 259

5. Read both the verniers A and B. The reading at A will give the angle directly.

The reading at B will be 180° + –POQ.

6. If there is any difference, take the average of the two values as the correct

angle.

Horizontal angles are measured this way for ordinary work. The accuracy can be

improved by reading the angles with face-left and face-right observations and taking

the average of the two. For more precise work, the angles are repeatedly measured

with both the faces and the average taken. This method is known as the repetition method and is described below.

6.4.1 Method of Repetition In the method of repetition, the horizontal angle is measured a number of times

and the average value is taken. It is usual to limit the number of repetitions to three

with each face except in the case of very precise work. With large number of rep-

etitions, errors can also increase due to bisections, reading the verniers, etc. Very

large number of repetitions necessarily do not lead to a more precise value of the

angle. However, a number of errors are eliminated by the repetition method. The

procedure is as follows (Fig. 6.6).

P

Q

O

0°00

00¢

1st

61° 112 40¢ ≤

O

P

Q

2nd

123° 25 20¢ ≤

61°

4240

¢≤

3rdO

P

Q

123°

2520

¢≤

185° 08 20¢ ≤(a) (b) (c)

Fig. 6.6 Repetition method

1. Angle POQ is to be measured. Set up, centre, and level the theodolite at O.

Ensure that the instrument is in the normal position, i.e., face left.

2. Set the instrument to read 0° 00¢ 00≤. For this release the upper clamp and

bring the zero of the vernier (at vernier A) very close to the zero of the circle.

Clamp the upper plate and using the upper tangent screw, coincide the two

zeros exactly.

3. Loosen the lower clamp and rotate the instrument so that the left signal at P is

approximately bisected. Tighten the lower clamp and using the lower tangent

screw, bisect the signal at P exactly. Read the verniers at A and B. The reading

should not change and they should read zero and 180°.

4. Loosen the upper clamp and rotate the instrument clockwise to approximately

Q exactly.

5. Read the verniers at A and B. The reading at A gives the value of the angle

directly. The reading on the vernier at B will be 180° + the angle. Record both

the readings.

6. Release the lower clamp and rotate the instrument clockwise to bisect the sig-

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260 Fundamentals of Surveying and Levelling

exactly. Check the readings on the verniers at A and B. They should remain

the same as recorded.

7. Release the upper clamp and rotate the instrument to bisect the signal

approximately. Lock the upper clamp and bisect the signal exactly with the

upper tangent screw. Read the verniers A and B. The readings should be twice

the angle. Record the readings from both the verniers.

8. Repeat the procedure once more. We thus have three repetitions of the value.

Record the readings of both the verniers.

9. Change the face of the instrument. Invert the telescope and make it face right.

Repeat the above procedure to have three more readings of the angle.

10. The average of the face-left readings and the average of the face-right readings

are averaged to get a very precise value of the angle.

Sets for precision work In works requiring greater precision, sets of readings are taken. One set may consist

of six face-left readings and an equal number of face-right readings. The readings

may be taken in clockwise and counterclockwise directions with different faces.

A number of similar sets may be taken. Sets may also start with a different initial

¢ 00≤, and so on. The average

of the average readings from the various sets will give a very precise value of the

angle. Such sets are taken in triangulation work. The readings can be recorded as

shown in Table 6.1.

Table 6.1 Recording of observations by repetition method

Set 1

Instrument

at

Sight

to

Face/swing

repetition number

Reading at Mean reading

A B

Left/right ° ¢ ≤ ¢ ≤ ° ¢ ≤O P 0 00 00 00 00 0 00 00

Q 1 61 42 40 42 40 61 42 45

P 61 42 40 42 40

Q 2 123 25 20 25 20 123 25 20

P 123 25 40 26 40 123 26 40

Q 3 185 08 20 08 20 185 08 20

Mean value of the angle = 61° 42 ¢ 47≤Set 2

Instrument

at

Sight

to

Face/swing repeti-

tion number

Reading at Mean reading

A B

Right/left ° ¢ ≤ ¢ ≤ ° ¢ ≤O P 00 00 00 00 00

Q 1 61 42 40 42 40

P

Q 2 123 25 40 25 40

P

Q 3 185 08 00 08 00 185 08 00

Mean value of the angle = 61° 42¢ 40≤

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Theodolite Surveying 261

The method of repetition helps to eliminate the following errors.

(a) Errors caused by the eccentricity of the centres and verniers, by reading both

the verniers and averaging.

(b) Graduation errors by reading from different parts of the circle.

(c) Imperfect adjustment of the line of collimation and horizontal axis by face-left

and face-right observations.

(d) Observational errors and other errors tend to be compensated by the large

number of readings.

However, the errors due to levelling cannot be compensated. This has to be done

by permanent adjustment. Also a large number of repetitions tend to increase the

wear of clamp and tangent screws.

Therefore, from the two sets,

Mean value of the angle = (1/2)(61° 42¢ 47≤ + 61° 42¢ 40≤ ) = 61° 42¢ 44≤

6.4.2 Method of Reiteration The method of reiteration is

another method of measuring

horizontal angles. This method is

useful when a number of angles

are to be measured at one point. In

Fig. 6.7, let O be the point where

the instrument is set up and P, Q,

R, and S be the stations. Angle

POQ, QOR, and ROS are to

be measured. In the retiration

method, each angle is measured

of sight is brought back to P, i.e.,

the line of sight is made to close

the horizon. The instrument is

turned through 360°. Obviously, the instrument should read, upon closing the ho-

rizon, the same reading set initially at P. The procedure is as follows.

1. Set up and level the theodolite at O. Keep the instrument in the normal posi-

tion, i.e., face left. Set the vernier at A to read zero using the upper clamp and

upper tangent screw. Check that the vernier at B reads 180°.

2. Loosen the lower clamp and swing the instrument to bisect the station mark at

at P. Check that the verniers at A and B read zero and 180°, respectively.

3. Release the upper plate with the upper clamp, swing the instrument clockwise

to bisect the signal at Q. Tighten the clamp and using the upper tangent screw,

bisect the mark at Q exactly.

4. Read the verniers at A and B and record both the readings.

5. Release the upper clamp screw, bisect the signal at R. Tighten the clamp and

bisect the mark at R exactly with the upper tangent screw. Read the verniers at

A and B and record the readings.

6. Continue the procedure with other stations.

PQ

R

S

Closingthe horizon

0° 0000

¢≤

68°32

40¢≤

130° 13 00¢ ≤O

233°

0720¢

Fig. 6.7 Reiteration method

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262 Fundamentals of Surveying and Levelling

7. After the last angle has been recorded, release the upper clamp and swing the

instrument to close the horizon and bisect the station mark at P. Check that the

verniers at A and B now read the initial reading set while starting. If there is

the lower clamp and the lower tangent screw should not be touched during the entire process after bisecting the station mark at P.

9. Each angle is calculated from the mean value of the readings of verniers A and

B at each station. The difference between these mean values gives the value of

the required angle.

an inverted position (right face). To be more accurate, the second set of

readings may be taken with the initial value of the reading different from zero.

The instrument can be set to read 30° or 60° initially. The reading may also be

taken in an counterclockwise direction along with the change of face. A second

average value can be obtained.

the angle.

As in the repetition method, sets of values can be obtained and averaged for more

precise work. The readings can be recorded as shown in Table 6.2.

Table 6.2 Recording of angles by the reiteration method

Face left/right swing

Instrument

at

Sight

to

Reading at Mean reading Angle Value

A B

° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤O P 00 00 00 00 00 00 00 00

Q 68 32 40 32 40 68 32 40 POQ 68 32 40

R 130 13 00 13 00 130 13 00 QOR 61 40 20

S 233 07 20 07 20 233 07 20 ROS 102 54 20

P 00 00 00 00 00 00 00 00

Face right/right swing

Instru-

ment at

Sight

to

Reading at Mean read-

ing

Angle Value

A B

° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤O P 00 00 00 00 00 00 00 00

Q 68 32 20 32 20 68 32 20 POQ 68 32 20

R 130 13 20 13 20 130 13 20 QOR 61 41 00

S 233 07 40 07 40 233 07 40 ROS 102 54 20

P 00 00 00 00 00 00 00 00

The mean values of the angles are

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Theodolite Surveying 263

–POQ = 68° 32¢ 30≤, –QOR = 61° 41¢ 10≤, –ROS = 102° 54¢ 20≤

6.5 Measuring Vertical Angles

A vertical angle is made by an inclined line of sight with the horizontal. The line

of sight may be inclined upwards or downwards from the horizontal. Thus one

may have an angle of elevation or depression. See Fig. 6.8. For measuring vertical

angles, the theodolite is levelled with respect to the altitude bubble.

6° 52 40¢ ≤

12° 32 40¢ ≤

8° 46 20¢ ≤

P

Q

R

Horizontal line

Fig. 6.8 Measuring vertical angles

The procedure for measuring vertical angles is as follows.

1. Set up the theodolite at the station from where the vertical angle is to be mea-

sured. Level the instrument with reference to the plate bubble.

arm. This bubble is generally more sensitive. The procedure for levelling is

the same. Bring the altitude level parallel to two foot screws and level till the

bubble traverses. Swing through 90° to centre the bubble again with the third

foot screw. Repeat till the bubble traverses.

3. Swing the telescope to approximately direct the line of sight towards the signal

at P. Loosen the vertical circle clamp screw and incline the line of sight to bisect

P. Clamp the vertical circle and bisect the signal exactly with the horizontal

cross hair.

4. Read the verniers C and D. The average of these readings gives the value of

the angle.

This procedure assumes that the instrument is properly adjusted. If there is an index

error, the instrument does not read zero when the bubble is in the centre and the

line of sight is horizontal, the adjustment is done by the clip screw. There may be

a small index error, which can be accounted for in the value of angle. The readings

can be recorded as shown in Table 6.3.

6.5.1 Measuring Vertical Angle Between Two Points The two points may be above the horizontal or below the horizontal or one may be

above and the other below. In all cases, the vertical angles between the instrument

and the points are measured. If the points lie on the same side of the horizontal, the

vertical angle between the points is the difference between the measured angles. If

they lie on either side of the horizontal through the instrument, the vertical angle

between the points is the sum of the angles measured.

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264 Fundamentals of Surveying and Levelling

Table 6.3 Recording of vertical angles

Face left

Instrument at Sight

to

Reading on vernier Mean angle Vertical angle

C D

° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤O P 12 32 40 32 40 12 32 40

Q 8 46 20 46 20 8 46 20 3 46 20

R (–)6 52 40 52 40 (–)6 52 40 19 25 20

Face right

Instrument at Sight

to

Reading on vernier Mean angle Vertical angle

C D

° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤O P 12 32 20 32 20 12 32 20

Q 8 46 20 46 20 8 46 20 3 46 00

R (–) 6 52 20 52 20 (–)6 52 20 19 24 40

The mean vertical angles are:

between P and Q = 3° 46¢ 10≤, between P and R = 19° 25¢ 00≤

6.6 Other Theodolite Operations

The theodolite is a very versatile instrument and can be used effectively for many

accurate operations. Some of these are discussed below. Traversing with the the-

odolite is discussed in Section 6.7.

6.6.1 Measuring the Magnetic Bearing of a LineThe following procedure is adopted to

measure the magnetic bearing of a line.

1. To determine the magnetic bearing of

line PQ, set and level the theodolite at

P (Fig. 6.9).

release the upper plate and swing the

instrument to set the reading on vernier A to read zero approximately. Clamp

the upper plate and with the upper tangent screw, make the zero of the vernier

and the circle coincide exactly.

3. Release the lower plate and the magnetic needle of the compass. Swing the

instrument so that the magnetic needle is nearly at the centre of its run, with the

north end of the needle pointing to the zero of the graduations on the compass.

Tighten the lower clamp and using the lower tangent screw, bring the magnetic

N

Q

P

Magnetic

bearing

Fig. 6.9 Measuring magnetic bearing

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Theodolite Surveying 265

needle to read exactly zero at the north end. Check the verniers at A and B. They

should read zero and 180° as set earlier.

4. The line of sight of the instrument is in the direction of the magnetic meridian

at P. Release the upper clamp and swing the instrument to bisect the signal at

tangent screw, bisect the signal exactly at Q.

5. Read both the verniers. The average of the two readings gives the magnetic

bearing of the line PQ.

6.6.2 Prolonging a Given Line Let PQ be the line given. It is desired to prolong the line to another point T in line

with it and establish intermediate points R and S (Fig. 6.10). There are many ways

to do this. The following three methods are common.

PQ R S

T

Fig. 6.10 Prolonging a line

P

Q RS

T

Fig. 6.11 Back sighting method

Direct method The procedure is as follows.

1. Set up and level the theodolite at P.

2. Swing the instrument and bisect the signal at Q. Exact bisection can be done

with the lower clamp and the lower tangent screw.

3. Establish a point along the line of sight at R.

4. Shift the instrument to Q. Level the instrument and sight the signal or station

mark at R. Establish a third point at S along the line of sight.

5. Shift the instrument and set it up at R. Sight the station mark at S. Establish

another station along the line of sight.

6. Continue the process until the point T is reached.

The method is suitable when the instrument is aligned properly.

Back sightingThe following procedure is adopted (Fig. 6.11).

1. Set up and level the theodolite at Q. Swing the instrument to take a back sight

on P. Clamp both the motions (upper and lower plates) and transit the tele-

scope.

2. Set a point R along the line of sight at a convenient distance.

3. Shift the instrument to R. Set up and level the instrument at R. Swing the instru-

ment and take a back sight on the station mark at Q. Clamp both the motions

and transit the telescope. Fix a point S along the line of sight.

reached.

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266 Fundamentals of Surveying and Levelling

This method also assumes that the instrument is properly aligned. If not, when the

instrument is transited, the line of sight may not be along PQ and dotted lines will

be the result.

Double sightingThe double sighting method is used when the instrument is not properly aligned

and does not give precise results. The procedure is as follows (Fig. 6.12).

1. Set up the theodolite at Q and level the theodolite. Release the lower plate,

swing the instrument, and take a back sight on P.

2. With both the plates clamped, transit the telescope and establish a point in line

with the line of sight.

3. Change the face of the instrument and again take a back sight on P.

4. With both motions clamped, transit the telescope. The line of sight should now

bisect the station mark established earlier. If not, mark a second point in line

with the earlier point.

5. Point R¢ was established with the left face and point R≤ is established with the

right face. Take the midpoint of R¢ and R≤ and mark this point R.

6. Shift the instrument to R and repeat the process of double sighting to establish

other points till the point T is reached.

PQ R S

T

R≤

Fig. 6.12 Double sighting

Laying a straight line between two given points There are three methods for laying a straight line between two given points depend-

ing upon whether the two stations are intervisible, intervisible from an intermediate

point, or are not intervisible from any intermediate point.

When the stations are intervisible When the two given stations are intervisible,

the procedure is simple, as follows (Fig. 6.13).

1. Set up and level the theodolite at P.

2. Sight the signal at Q and clamp both the motions. Finely bisect the signal at Q

using the lower tangent screw.

4. PRSTQ is a straight line.

P

R S T

Q

Fig. 6.13 Laying out a straight line

When the stations are visible only from an intermediate point In this case a

method known as balancing-in is used. The procedure is as follows (Fig. 6.14).

1. Determine a point midway between P and Q from which both P and Q are vis-

ible.

2. By judgement, establish a point R in line with P and Q.

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Theodolite Surveying 267

3. Set up and level the theodolite at R.

4. Swing the instrument and take a back sight on P. Transit the telescope and sight

the station mark at Q. It may not be in line as the position of R is only set by

estimation.

5. Estimate a new position of R to be in line with P and Q.

6. Repeat the procedure till, after back sighting on P and plunging the telescope,

the signal at Q is bisected.

7. R is then a point in line with P and Q.

P Q

R1

R2

RP Q

Fig. 6.14 Balancing-in method

When the stations are not visible from any intermediate point In this case the

random line method is used. The procedure is as follows (Fig. 6.15).

1. Run a random line at any angle to one side of the line joining the two stations

P and Q.

2. To do this, set up the theodolite at P. Centre and level the instrument.

3. Take a line of sight along PQ¢ and establish point Q¢ such that Q is visible from

Q¢ and the distance QQ¢ can be measured.

¢, S¢, etc. along the line of

sight of PQ¢.5. Measure QQ¢, PQ¢, PR¢, PS¢, etc. accurately with a steel tape.

6. Set up and level the instrument at Q¢. Measure angle PQ¢Q = a accurately.

7. Set up the instrument at R¢ and level it. Swing the instrument and lay off an angle

equal to a. The point R on line PQ can be obtained by measuring a distance R¢Ron the line laid at angle a. (The method to lay off an angle is given later.)

8. The length of this line can be determined using similar triangles PQ¢Q and

PR¢R. R¢R is equal to QQ¢ ¥ PR¢/ PQ¢.9. Repeat the same procedure with points S¢, etc. PRSQ is a straight line between

P and Q.

PR

S T

S¢T¢

Q

Q¢a

Fig. 6.15 Random line method

Locating the point of intersection of two given lines

the point of intersection of the two lines. The procedure is as follows.

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268 Fundamentals of Surveying and Levelling

1. Set up and level the theodolite at P.

Release the lower clamp, swing the

instrument, and bisect the signal at

exactly bisect the station mark at Q.

2. Along the line of sight, set up two

points 1 and 2, roughly on either

side of an estimated intersection

point of PQ and RS.

3. Set up and level the instrument at

R. Release the lower clamp, swing

the instrument to sight S. Clamp and

using the lower clamp screw, exactly bisect the station mark at S.

4. The intersection point O lies along this line of sight and between the points 1

and 2 established earlier.

sighting from R to S, obtain the point O and mark it accurately.

6. The point can be checked by setting up the instrument at O and checking

whether PQ and RS are in line with the station point.

Laying off a given angle The procedure for setting an angle on the ground is as follows (Fig. 6.17).

1. Let the angle to be set be 38° 47¢ 20≤ with

a line PQ.

2. Set up the theodolite at P, centre it, and level

it with reference to the plate level.

3. Release the upper clamp, swing the instru-

ment to make vernier A read 0° 00¢ 00≤. Set

the reading exactly to zero with the upper

tangent screw. Read vernier B, which should register 180° 00¢ 00≤.

4. Release the lower clamp and swing the instrument to bisect the signal at Q.

Exactly bisect the signal at Q with the lower tangent screw. Check the reading

on both the verniers, which should be the same as set before (0° 00¢ 00≤).

5. Release the upper clamp and swing the instrument clockwise. Set vernier A to

read approximately the given angle and clamp. Set the angle to the given value

exactly using the upper tangent screw. Check the angle on vernier B.

6. Set a point R along the line of sight. Angle QPR is the given angle.

Laying off a given angle by repetitionIf greater precision is required in setting the angle, the method of repetition can be

used. While a given angle can be measured by the method of repetition directly,

there is no corresponding direct method to set an angle. An indirect method is

employed. The procedure is as follows (Fig. 6.18).

1. Let the angle to be set off be 54° 36¢ 35≤ with the line OP.

2. Set up and level the theodolite at O. Set the instrument to read zero using the

upper clamp and the upper tangent screw. Check vernier B also.

P R

Q

38° 47 20¢ ≤

0°00

00¢≤

Fig. 6.17 Laying out an angle

O

1

2

S

P

Q

R

Fig. 6.16 Intersection point of two lines

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Theodolite Surveying 269

3. Release the lower clamp, bisect the station

mark at P, and exactly bisect the mark with

the lower tangent screw. Check the readings

on both the verniers; they should remain the

same.

4. Release the upper clamp, swing the instrument

clockwise, and set the instrument to read 54°

36¢ 00≤. With the upper tangent screw, make

the readings exactly 54° 36¢ 00≤. Make a sta-

tion mark along the line of sight. Point Q¢ is

established.

5. Measure the angle POQ¢ by the repetition method. Take three readings with

face left and three readings with face right. Find the average value of the angle

from these observations. Let the average value be equal to 54° 36¢ 12≤.

6. The difference between the angle to be set and this angle is 23≤. The angle is

corrected by linear measurements, as the difference in angles is too small.

7. Let the length of the line OQ¢ be 100 m. Then Q¢Q = 100 tan (23≤ ) = 0.011 m.

8. Measure this distance accurately along the perpendicular with a steel tape to

get point Q. Angle POQ is the required angle.

9. Measure the angle POQ again by repetition to check.

6.7 Theodolite Traversing

We have seen that chain traversing or chain triangulation is done using only a chain

and the de -

ing can be done with the directions of lines coming from bearings and distances

measured with a chain or tape. With the theodolite, traversing can be done by many

angles, or external angles of a traverse, with the distances measured using a tape.

Open and closed traverses can be surveyed accurately with a theodolite.

of details with offsets. The skeleton is not restricted to a framework of triangles

and can have any shape. The survey lines are arranged to suit the terrain and the

details to be located.

The basic procedure for theodolite traversing is the same as that in any other

method of traversing. First reconnaissance has to be conducted with a sketch drawn

of the terrain using approximate locations of traverse stations, then the important

details are to be picked up, the intervisibility of stations to be checked, and the basic

equipment required for the survey to be collected. Theodolite traversing requires

station marking tools such as pegs, arrows, a theodolite with its stand, and a steel

tape. The decision regarding which method of surveying will be used, by angles

or bearings, should also be taken in advance including the scale of the plot to be

6.7.1 Traversing with Magnetic BearingsThe method of traversing with magnetic bearings is suitable in places where local

measuring angles. This method is also known as the fast needle method of traversing.

P

OQ

Q¢54° 36 00¢ ≤

Fig. 6.18 Laying out an angleby repetition

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270 Fundamentals of Surveying and Levelling

Direct method (without transiting)In the direct method, the bearings at

each station are taken by swinging the

telescope. The procedure is as follows

(Fig. 6.19).

1. Set up and level the theodolite at

station P of the traverse PQRSTP, a

closed traverse.

tangent screw, set vernier A to read

zero.

3. Loosen the magnetic needle. Re-

lease the lower clamp and point

the telescope in the direction of the

magnetic meridian till the magnetic

needle comes to rest at the zero posi-

the north end of the magnetic needle

to read exactly zero.

4. Release the upper plate and swing the instrument to bisect the signal at Q. With

the upper tangent screw, bisect the station mark exactly.

5. Read vernier A, which gives the bearing of the line PQ.

6. Keeping both the clamps tight, shift the instrument to Q. Set up and level the in-

strument. Check the reading on vernier A. It should be the same as the magnetic

bearing of the line PQ (if not, this can be corrected and the bearing value noted

earlier be set on vernier A).

7. Release the upper clamp. Swing the instrument clockwise to bisect the station

Read the vernier at A and note down the reading.

8. With both clamps tight, shift the instrument to R and repeat the procedure. The

work is continued at all stations in a similar manner.

The following points should be carefully noted.

(a) We get the correct reading of the bearing of PQ at P on vernier A.

(b) At Q, as the theodolite is swung to sight R, the instrument is 180° out of ori-

entation and a correction has to be applied to the reading on vernier A. In this

position, vernier B gives the correct reading of the magnetic bearing of line

QR. This can be seen from the sample readings given in Table 6.4.

(c) It is more convenient to read one vernier only throughout the operation. In such

a case, the bearings at Q and S (or the second, fourth, sixth, etc.) stations need

etc.) can be read directly on vernier A.

(d) Alternately, read vernier A for the magnetic bearings of PQ, RS, and TP (or

ST (or the second, fourth, sixth, etc.).

N

N

N

N

N

S

T

P

Q

R

Fig. 6.19 Magnetic bearings–Direct method (without transiting)

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Theodolite Surveying 271

before measuring the bearing of PQ. This can be used for checking.

Table 6.4 Recording magnetic bearings

Instrument

at

Sight to Reading on

vernier A

Reading on

vernier B

Line Bearing

° ¢ ≤ ° ¢ ≤ ° ¢ ≤P 00 00 00 00 00 00

Q 62 42 20 242 42 20 PQ 62 42 20

T 201 59 40 21 59 40

Q R 334 53 40 154 53 40 QR 154 53 40

R S 202 32 40 22 32 40 RS 202 32 40

S T 101 43 20 281 43 20 ST 281 43 20

T P 21 59 40 201 59 40 TP 21 59 40

Direct method (with transiting)In this method, instead of swinging the

instrument, the telescope is transited at

the second station. This gives the correct

reading of the bearings at the station. The

procedure is as follows (Fig. 6.20).

1. Set up and level the theodolite at P. Set

the reading on the vernier to zero with

the upper clamp and the upper tangent

screw.

2. Release the lower clamp and swing the

instrument towards the magnetic merid-

ian. Release the needle of the compass.

With the lower tangent screw, set the

magnetic needle to read exactly zero.

3. Release the upper clamp and swing the instrument to bisect the signal at Q.

With the upper tangent screw, exactly bisect the station mark Q. Read vernier

A and record the reading.

4. With both plates clamped, shift the instrument to Q, and set up and level the

instrument.

5. Release the lower clamp and swing the instrument to take a back sight on P.

Exactly bisect the signal with the lower tangent screw. Check that the reading

on vernier A is the bearing of PQ.

6. Transit the telescope. The line of sight is now directed in the direction of the

extension of line PQ. The reading of the bearing is correct in this position.

7. Release the upper clamp and swing the instrument to bisect the station mark at

R. Exactly bisect the signal with the upper tangent screw. Read vernier A and

record. This will be the correct bearing of line QR.

N

P

TS

RQ

Fig. 6.20 Magnetic bearings–Direct method (with transiting)Prev

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272 Fundamentals of Surveying and Levelling

8. Shift the instrument to the next station and repeat the procedure.

The following point may be noted in this method: the instrument is transited at

every station to orient it with the line of sight.

Back bearing methodIn this method the instrument is set to the back bearing of the line to get the correct

orientation of the line. The procedure is as follows.

1. Set up and level the theodolite at P. With the upper clamp and upper tangent

screw, set vernier A to read zero. With the lower clamp and lower tangent screw,

bring the line of sight to the magnetic meridian by releasing the magnetic needle.

When the needle reads zero, clamp the instrument.

2. Release the upper clamp, swing the instrument to bisect the signal at Q, and

bisect the signal exactly with the upper tangent screw. Read vernier A and

record the reading as the bearing of PQ.

3. Shift the instrument to Q. Calculate the back bearing of line PQ. With the up-

per clamp and the upper tangent screw, set the reading of vernier A to the back

bearing of PQ. Release the lower clamp and take a back sight on P. With the

oriented.

4. Release the upper clamp, swing the instrument, and bisect the signal at R. With

the upper tangent screw, exactly bisect the signal. Read vernier A. This reading

gives the bearing of line QR.

5. Shift the instrument to other stations and repeat the procedure.

The following points may be noted in this procedure.

(a) The theodolite is not transited at any station. Thus the error due to line of sight

reversal is not present.

(b) The instrument is brought into correct orientation by setting the back bearing

of the previous line.

(c) The instrument is swung and rotated in a horizontal plane at every station to

bisect the signal at the next station.

(d) The reading on vernier A gives the correct bearings of the lines at every sta-

tion.

6.7.2 Traversing by AnglesInstead of bearings of lines, the angles of a traverse can be measured with the

theodolite. The angles measured can be included (interior) angles, exterior angles,

-

measured to get the direction of the magnetic meridian at that place.

Traversing by interior or exterior anglesFigure 6.21 shows the interior and exterior angles of a traverse. If the traverse is

covered in the clockwise direction, the angles measured are interior angles. If we

go counterclockwise over the traverse, we measure the exterior angles. The general

procedure in either case is as follows.

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Theodolite Surveying 273

Q

R

S

T

P

Q

P

T

S

R

(a) Interior angles (b) Exterior angles

Fig. 6.21 Traversing by angles

Table 6.5 Recording of interior angles

Face left/swing right; traversing clockwise

Instrument at Sight to Reading on vernier Angle Horizontal angle

A B° ¢ ≤ ¢ ≤ ° ¢ ≤

P Q 00 00 00 00 00

T 139 17 20 17 20 TPQ 139 17 20

Q R 00 00 00 00 00

P 87 48 40 48 40 PQR 87 48 40

R S 00 00 00 00 00

Q 132 21 00 21 00 QRS 132 21 00

S T 00 00 00 00 00

R 100 49 00 49 00 RST 100 49 00

T S 00 00 00 00 00

P 79 44 00 44 00 STP 79 44 00

Face right/swing right; traversing clockwise

Instrument at Sight

to

Reading on vernier Angle Horizontal

angle

Mean angle

A B

° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤P T 00 00 00 00 00

Q 139 17 00 17 00 TPQ 139 17 00 139 17 10

Q P 00 00 00 00 00

R 87 48 20 48 20 PQR 87 48 20 87 48 30

R S 00 00 00 00 00

Q 132 21 20 21 20 QRS 132 21 20 132 21 10

S T 00 00 00 00 00

R 100 49 20 49 20 RST 100 49 20 100 49 10

T P 00 00 00 00 00

S 79 43 40 43 40 STP 79 43 40 79 43 50

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274 Fundamentals of Surveying and Levelling

1. Set up and level the theodolite at the starting station P. Measure the magnetic

bearing of line PQ as per the procedure outlined earlier. With the upper clamp

and the upper tangent screw, set the instrument to read zero on vernier A. Check

the reading on vernier B. Record both the readings.

2. Release the lower clamp and swing the instrument to sight the station mark Q.

Bisect the signal exactly with the lower tangent screw. Check that the reading

on both the verniers has not changed. Release the upper clamp and swing the

instrument to bisect the station mark at T. Exactly bisect the signal at T with

the upper tangent screw. Read verniers A and B and record the observations.

This gives the interior angle at station P.

3. Shift the instrument to Q, set it up, and level it. Set the vernier readings to zero

again with the upper clamp and the upper tangent screw. With the lower clamp

and lower tangent screw, bisect exactly the station mark at R. Check the read-

ings on the verniers, which should read zero and 180°.

4. Release the upper clamp and swing the instrument to bisect the signal at P. Read

verniers A and B and record the readings. This will give the interior angle at Q.

5. Repeat the procedure at all stations of the traverse.

The readings can be recorded as shown in Table 6.5. To achieve greater accuracy,

the angles can be measured at a station with both faces, face left and face right.

For high, precision work, angles can be read by the method of repetition with three

repetitions for each face. When you go counterclockwise and measure the angles

from the previous station to the forward station, you measure the exterior angles.

The procedure is exactly the same.

Traversing by deflection angles is the

preferred method for open traverses.

The method can be used for closed tra-

verses as well. The procedure is as follows

(Fig. 6.22).

1. For the closed traverse PQRSTP

shown, set up and level the theodolite

at P. Observe and record the magnetic

bearing of the line PQ by the method

outlined above.

tangent screw, set vernier A to read

zero. Check vernier B. Record the

readings of both the verniers.

-

gent screw, bisect the signal at the last station T. Transit the telescope. Release

the upper clamp, and swing the instrument to bisect the signal at the next station

Q. With the upper tangent screw, bisect the signal exactly. Read the verniers.

bearing, this can help as a check.

Q

R

ST

P

Fig. 6.22angles

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Theodolite Surveying 275

4. Shift the instrument to Q, set it up, and level it. With the upper clamp and tangent

screw, set vernier A to read zero. With the instrument reading zero, release the

lower clamp and swing the instrument to bisect the signal at P. Exactly bisect

the signal with the lower tangent screw.

5. Transit the telescope. Release the upper clamp and swing the instrument to

bisect the signal at R. Clamp the upper plate and, with the upper tangent screw,

exactly bisect the signal. Read both the verniers and record.

6. Repeat the procedure at all the stations.

The observations can be recorded as shown in Table 6.6. It should be noted that

right, when the next line is to the right of the continuation of the previous line. A

previous line. It should be recorded simultaneously with the observed angles.

Table 6.6

Face left

Instrument

at

Sight to Reading on vernier Mean value

A B° ¢ ≤ ¢ ≤ ° ¢ ≤ ° ¢ ≤

P 00 00 00 00 00

Q 61 46 20 46 20 61 46 20

Q P 00 00 00 00 00

R (R) 15 36 40 36 20 15 36 30 15 36 30

R Q 00 00 00 00 00

S (R) 44 32 20 36 40 44 36 30 44 36 40

S R 00 00 00 00 00

T (R) 54 28 40 28 40 54 28 40 54 28 40

P

Q

R

S

T

N

Fig. 6.23 Open traverse–Direct angles

Open traverse with direct angles

open traverse. The procedure is the following (Fig. 6.23).

1. Set up the theodolite at station P and level it. Record the magnetic bearing of

line PQ.

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276 Fundamentals of Surveying and Levelling

2. Shift the instrument to Q. Set up and level it. Set vernier A to read zero using

the upper clamp and upper tangent screw. Check vernier B also.

3. Release the lower clamp and swing the instrument to take a sight on the station

mark at P. Exactly bisect the signal with the lower tangent screw.

4. Release the upper plate with the upper clamp and swing the instrument clock-

wise to bisect the signal at R. This reading gives the angle between the lines

QP and QR.

5. Shift the instrument to other stations and repeat the procedure.

Closed traverse with central angles In the case of a closed traverse, another

procedure can be adopted to survey the

skeleton of the traverse. The method

of reiteration can be conveniently em-

ployed here. The procedure is as follows

(Fig. 6.24).

1. Select a convenient point O within the

traverse from which all the points of

the traverse are visible and chaining

to those points is possible.

2. Set up and level the theodolite at O.

Set the instrument to read zero with the

upper clamp and upper tangent screw.

Vernier A should read zero and vernier B should read 180°.

Exactly bisect it with the lower tangent screw.

4. Check the reading on the verniers. They should be the same as set. Release

the upper clamp and swing the instrument to bisect the signal at Q. Bisect it

5. Similarly, take sights to stations R, S, and T and note the angle readings from

both the verniers. Finally close the horizon by taking a sight to station P. Check

the reading on the verniers. They should be the readings we started with at P,

in this case zero and 180°.

6. Accuracy can be improved with a second set of readings with changed face,

starting from a reading other than zero at the start and reading angles in a

counterclockwise direction.

The readings can be recorded as shown in Table 6.7. The following points should

be noted.

(a) Along with the angles, the distances OP, OQ, etc. are measured with a tape.

These are required for plotting the traverse.

(b) As a check, the distances PQ, QR, etc. are also measured. This is not extra work,

as these distances are measured for the purpose of taking in details with offsets.

(c) There is no direct measurement of the traverse angles. They can be calculated

from the properties of triangles, knowing the three sides.

T

P

Q

S

R

O

Fig. 6.24 Closed traverse bycentral angles

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Theodolite Surveying 277

Table 6.7 Recording of observations

Face left

Instrument at Sight to Reading on vernier Angle Horizontal angle

A B

° ¢ ≤ ¢ ≤ ° ¢ ≤O P 00 00 00 00 00

Q 62 43 40 43 40 POQ 62 43 40

R 120 38 00 38 00 QOR 57 54 20

S 192 53 40 53 40 ROS 72 15 40

T 273 15 00 15 20 SOT 80 21 10

P 359 59 40 59 40 TOP 86 44 50

6.7.3 Interconversion of Angles

The theodolite measures the whole circle bearings of lines. These can be converted

to reduced bearings by the methods discussed in Chapter 3. Also, one can calculate

included angles from bearings and vice versa. Included angles can also be calculated

The following relationships of the angles of a closed traverse are known from

geometry:

(a) sum of the interior angles = (2n – 4) right angles

(b) sum of exterior angles = (2n + 4) right angles

It is desirable to draw a rough sketch of the traverse before attempting to solve

problems. The following examples illustrate these principles.

Example 6.1 The whole circle bearings of the lines of a closed traverse are given below.

Find the included angles of the traverse.

Line AB BC CD DA

Bearing 78° 40¢ 20≤ 152° 31¢ 40≤ 251° 18¢ 40≤ 3° 44¢ 15≤Solution It is desirable to draw a rough sketch of the traverse as shown in Fig. 6.25. The

solution is shown in Table 6.8.

Table 6.8 Solution to Example 6.1

Line Fore bearing Back bearing Angle at ° ¢ ≤AB 78° 40¢ 20≤ 258° 40¢ 20≤ A 105 03 55

BC 152° 31¢ 40≤ 332° 31¢ 40≤ B 106 08 40

CD 251° 18¢ 40≤ 71° 18¢ 40≤ C 81 13 00

DA 3° 44¢ 15≤ 183° 44¢ 15≤ D 67 34 25

First we calculate the back bearings of the lines. The included angle is the difference between

the back bearing of a line and the fore bearing of the next line. Thus,

–DAB = bearing of DA – bearing of AB = 183° 44¢ 15≤ – 78° 40¢ 20≤= 105° 03¢ 55≤

–ABC = bearing of BA – bearing of BC = 258° 40¢ 20≤ – 152° 31¢ 40≤= 106° 08¢ 40≤

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278 Fundamentals of Surveying and Levelling

–BCD = Bearing of CB – bearing of CD = 332° 31¢ 40≤ – 251° 18¢ 40≤= 81° 13¢ 00≤

–CDA = bearing of CD – bearing of DA = 71° 18¢ 40≤ – 3° 44¢ 15≤= 67° 34¢ 25≤

Check: Sum of included angles of a traverse = (2n – 4) right angles = 360 or a four-sided

C

251° 18 40¢ ≤

D3° 44 15¢ ≤

A

78° 40 20¢ ≤N B 152° 31 40¢ ≤

Q87° 48 40¢ ≤

132° 21 00¢ ≤ R

100° 49 00¢ ≤ST

79° 44 00¢ ≤

139° 17 20¢ ≤P

62° 42 20¢ ≤N

Fig. 6.25 Traverse of Example 6.1 Fig. 6.26 Traverse of Example 6.2

Example 6.2 ¢ 20≤. The

included angles of the traverse are given below. Calculate the whole circle bear-

ings of the lines.Station P Q R S T

Angle 139° 17¢ 20≤ 87° 48¢ 40≤ 132° 21¢ 00≤ 100° 49¢ 00≤ 79° 44¢ 00≤Solution A rough sketch of the traverse is shown in Fig. 6.26. The solution is given in

Table 6.9.

Table 6.9 Solution to Example 6.2

Station Included angle Line Fore bearing Back bearing

P 139° 17¢ 20≤ PQ 62° 42¢ 20≤ 242° 42¢ 20≤Q 87° 48¢ 40≤ QR 154° 53¢ 40≤ 334° 53¢ 40≤R 132° 21¢ 00≤ RS 202° 32¢ 40≤ 22° 32¢ 40≤S 100° 49¢ 00≤ ST 281° 43¢ 40≤ 101° 43¢ 40≤T 79° 44¢ 00≤ TP 21° 59¢ 40≤ 201° 59¢ 40≤

Example 6.3 A closed traverse ABCDEA was conducted and the included angles mea-

¢ 20≤vertex and the bearings of the lines.

Station A B C D

Angle 105° 03¢ 55≤ 106° 08¢ 40≤ 81° 13¢ 00≤ 67° 34¢ 25≤Solution A rough sketch of the traverse is shown in Fig. 6.27. The solution is given in

Table 6.10.

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Theodolite Surveying 279

Table 6.10 Solution to Example 6.3

Station Included angle Line Bearing

° ¢ ≤ ° ¢ ≤ ° ¢ ≤A 105 03 55 74 56 05 AB 78 40 20

B 106 08 40 73 51 20 BC 152 31 40

C 81 13 00 98 47 00 CD 251 18 40

D 67 34 25 112 25 35 DA 3 44 15

Example 6.4measured in a closed traverse ABCDEA

are given below. Find the included

angles of the traverse.

A 81° 32¢ 40≤B 65° 54¢ 20≤C 72° 14¢ 30≤D 68° 25¢ 50≤E

Solutioneither has not been measured or is not

in a closed traverse is equal to 360°,

¢20≤ ¢ 20≤. The included angle is equal to 180° – de-

A 81° 32¢ 40≤ 98° 27¢ 20≤B 65° 54¢ 20≤ 114° 05¢ 40≤C 72° 14¢ 30≤ 107° 45¢ 30≤D 68° 25¢ 50≤ 111° 34¢ 10≤E 71° 52¢ 40≤ 108° 07¢ 20≤

As a check, sum of the included angles = 540° 00¢ 00≤.

Example 6.5 A traverse was conducted and the exterior angles were measured. Find the

Station A B C D

Exterior angle 254° 56¢ 05≤ 253° 51¢ 20≤ 278° 47¢ 00≤Solution The sum of the exterior angles of a closed traverse = (2n + 4) right angles. This

condition can be used to evaluate the unknown exterior angle.

(2n + 4) right angles = (2 ¥ 4 + 4) ¥ 90 = 12 ¥ 90° = 1080°

Sum of given angles = 787° 34¢ 25≤

Exterior angle at D = 1080° 00¢ 00≤ – 787° 34¢ 25≤ = 292° 25¢ 35≤

A: 74° 56¢ 05≤, B: 73° 51¢ 20≤, C: 98° 47¢ 00≤, D: 112° 25¢ 35≤

Example 6.6

¢ 30≤.

67° 34 25¢ ≤D

105° 03 55¢ ≤A

78° 40 20¢ ≤N

106° 08 40¢ ≤B

81° 13 00¢ ≤C

Fig. 6.27 Traverse of Example 4.3

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280 Fundamentals of Surveying and Levelling

P 62° 44¢ 20≤Q 48° 22¢ 00≤R 57° 36¢ 40≤S 71° 15¢ 20≤T 120° 01¢ 40≤

Solution The solution is shown in Table 6.11.

Table 6.11 Solution to Example 6.6

Station Line Bearing° ¢ ≤ ° ¢ ≤

P 62 44 20 PQ 54 25 30Q 48 22 00 QR 102 47 30R 57 36 40 RS 160 24 10S 71 15 20 ST 231 39 30T 120 01 40 TP 351 41 10

6.8 Locating Landscape Details with the Theodolite

We have discussed so far methods to survey the main frame or the skeleton of the

survey. In most surveys, it is necessary to locate details such as buildings, railway

lines, canals, and other landmarks along with the survey. A transit with a steel tape

is used to locate details, and many methods are available, as the transit is an angle-

measuring instrument. The following methods can be used.

Angle and distance from a single station A point can be located with an angle to the station along with the distance from that

station as shown in Fig. 6.28(a). The angle is preferably measured from the same

reference line to avoid confusion. A sketch with the line and the distance and angle

measured will help in plotting later. A road can be located as shown in Fig. 6.28(b).

Angles to a number of points are measured and with each angle two distances are

measured to locate the road.

Angle from one station and distance from anotherIf for any reason, it is not possible to measure the angle and distance to an object

from the same point, it may be possible to locate the point by measuring angles

from one station and distances from the other. The recorded data should clearly

indicate the stations from which the angle and distance are measured. Figure 6.28(c)

shows this method of measuring. The angle is measured from station A to point

P. When the instrument is shifted to B, the distance to point P is measured from B

with a steel tape.

Angles from two stationsIf for some reason, it is not possible to measure distances, then angles from two

stations are enough to locate a point. As shown in Fig. 6.28(d), the point P is located

by measuring angles to point P from stations A and B.

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Theodolite Surveying 281

Intermediate points on linesIf the stations are far away, any of the above methods can be used to locate details

from any one or two selected points on the main line. The angle and distance or two

angles from two different points can be used to locate points. The end of a chain

indicate the points and the angles and distances from the points.

OffsetsAs in the case of chain surveying, details can be located by offsets with the offset

rod or tape as described earlier from the main chain lines. This is suitable when a

number of points have to be located along the chain line.

6.9 Fundamental Lines and Desired Relationships

The theodolite has to be properly adjusted before it is used. All the methods

described above assume that the instrument is properly adjusted, though some

techniques whereby certain errors due to bad adjustment of the instrument can be

eliminated were discussed. However, the instrument should be in adjustment for

survey work. Here we discuss only the fundamental lines and their relationships for

getting accurate results using the theodolite. The actual procedure for adjustment

is discussed in Chapter 4.

P

d

q

q

(a)

O

(b)

P

d

BA A B

P

d

(c) (d)

(e)(f)

a b

a b g

Fig. 6.28 Filling-in details

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282 Fundamentals of Surveying and Levelling

The following are the fundamental lines.

1. The vertical axis

2. The horizontal or trunnion axis

3. The line of collimation or line of sight

4. Axis of altitude level

5. Axis of plate level

The meaning of these terms has been discussed earlier. The axes are shown in

Fig. 6.29. When the instrument is properly adjusted, the relationships between these

axes are the following.

(a) The horizontal axis must be perpendicular to the vertical axis.

(b) The axis of the plate level must be perpendicular to the vertical axis.

(c) The line of collimation must be at right angles to the horizontal axis.

(d) The axis of the altitude level (and telescope level) must be parallel to the line

of collimation.

(e) The vertical circle vernier must read zero when the line of sight is horizon-

tal.

Each one of these relations gives conditions for accurate measurement.

(a) When the horizontal axis is perpendicular to the vertical axis, the line of sight

generates a vertical plane when transited.

(b) When the axis of the plate level is perpendicular to the vertical axis, the vertical

axis will be truly vertical when the bubble traverses.

(c) When the line of collimation is at right angles to the horizontal axis, the tele-

scope when rotated about the horizontal axis will move in a vertical plane.

(d) When the line of collimation and the axis of altitude level are parallel, the

vertical angles will be measured without any index error.

(e) The index error due to the displacement of the vernier is eliminated when the

vernier reads zero with the line of collimation truly horizontal.

Altitude level

Bubble axis

Line of sight

Optical centre

Trunnion axis

Vertical axisCross hairs

Eyepiece

Vertical circle

Horizontal plate

Plate bubble

Axis of bubble

Objective

Fig. 6.29 Fundamental lines of a theodolite

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Theodolite Surveying 283

6.10 Errors in Theodolite Work

The theodolite is a very versatile but delicate instrument. The fundamental lines

retain the desired relationships just for a few uses. Thus, errors creep into the mea-

sured values. Other errors are due to incorrect use and operations by personnel.

Still other errors may be attributed to natural causes. These are discussed below.

6.10.1 Instrumental Errors

(i) Maladjustment of the plate level When the axis of the plate level, which is

a line tangential to the bubble tube at its centre, is not perpendicular to the

vertical axis, the horizontal circle is inclined when the bubble traverses. The

angles measured are not in a horizontal plane and an error will be introduced

in the measurements. This error becomes serious when the angle is measured

between two points at different elevations. A permanent adjustment has to be

carried out to set this right. The instrument can also be levelled with respect to

the altitude level, which is a more sensitive bubble tube.

(ii) Line of collimation not at right angles to the trunnion axis In this case, when

the telescope is transited (rotated in a vertical plane), it does not move in a

plane. The error can be eliminated by taking face right and face left observa-

tions and averaging the two.

(iii) Horizontal axis not perpendicular to the vertical axis When the horizontal axis

is not perpendicular to the vertical axis, an error is introduced, as the line of

sight will not move in a vertical plane but along an inclined plane. This error is

are at considerable difference in elevation. The error can be eliminated by both

face observations and averaging the result.

(iv) Eccentricity of inner and outer axis When the instrument is in adjustment, the

inner and outer axes coincide to form the vertical axis. If there is any difference

between them, the correct angle can be obtained by reading both the verniers

and averaging the result.

(v) Axis of altitude level not parallel to line of collimation This condition

results in error in the measurement of vertical angles, as the zero line of

the verniers does not act as a true reference line for measuring vertical

angles. Taking observations with both the faces and averaging the results

eliminate the error.

(vi) Faulty circle graduations The graduations of the circle may be faulty, not be-

ing equidistant in different parts of the circle. The error can be minimized by

taking the readings for an angle several times with different parts of the circle

and averaging the results.

(vii) Verniers being eccentric This condition arises when the zeroes of the verniers

are not at 180° or diametrically opposite. Taking readings at verniers A and B

and calculating the average eliminates this error.

6.10.2 Personal Errors

Personal errors arise due to faulty manipulation of the instrument controls and er-

rors in sighting, reading, and recording.

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284 Fundamentals of Surveying and Levelling

Inaccurate centring over a station

taken to see that the plumb bob is exactly over the station mark before taking the

reading. The effect of wrong centring can be derived as follows. In Fig. 6.30, let C

be the station mark. Due to inaccurate centring, the centre of the instrument is at

C1. In such a case, instead of the correct angle PCQ, the angle measured is PC1Q.

The correct angle PCQ is given by

–PCQ = PC1Q – a – b = PC1Q – (a + b )

aa

P P

C1

C

bb

Q Q

C1

C

(a) (b)

Fig. 6.30 Innacurate centring

Table 6.12 Variation of centring error with the length of sight

Centring error (mm) Length of sight (m) Angular error

5 25 41≤50 21≤

10 25 1¢ 22≤50 41≤

15 25 2¢50 1¢

20 25 2¢ 45≤50 1¢ 22≤

On the other hand, if the centring is done over C2 on the other side, then

Correct –PCQ = –PC2Q + (a + b )The error due to centring, ± (a + b ), depends upon the length of sights and the er-

ror in centring. Table 6.12 shows the error due to centring and how it varies wih the

length of sights.

Slip

error can also result from the clamps not being properly tightened or the shifting

head not being properly clamped.

Wrong manipulation of controlsThis is a common error. Clamping the instrument and operating the wrong tangent

screw can result in erroneous zero setting or reading of the angle. The remedy is to

familiarize oneself with the controls, operate them carefully, and verify the read-

ings frequently.

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Theodolite Surveying 285

Inaccurate levelling The levelling operation is fundamental to the theodolite. Levelling should be done

the instrumental error of the plate level not being perpendicular to the vertical axis.

Inaccurate bisection of targetAs angles are to be measured accurately, ideally a point target is required. Many

often, it is not possible to observe the cross mark on the peg over the station. Care

must be taken to ensure that if a ranging rod is sighted, the lowest point is observed

and an appropriate mark is made for bisection.

Non-verticality of ranging rodThe ranging rod or other object held over a station has to be truly vertical. Other-

wise an error is introduced in the reading. The error is inversely proportional to

the length of sight. The error is given by tan e = error in verticality/length of sight.

Displacement of target All stations should be properly marked. If the target is misplaced or shifted during

surveying, serious error can occur.

ParallaxTo avoid parallax, the eyepiece should be focused properly. The focusing of the

objective should be done at every sight to get a sharp and clear image.

Mistakes in setting and reading Mistakes in reading the main and vernier scales are common. These could be due

to reading the vernier in the wrong direction, inaccurate adjustment of the vernier,

or reading the main and vernier scales at an angle instead of directly over the

graduations.

6.10.3 Errors due to Natural Causes Errors due to natural causes include errors due to (i) settlement of the tripod due to soft soil

(ii) wind, causing vibrations and turning, and (iii) high temperature, causing faults

in reading due to refraction, differential expansion of different parts, and direct

6.11 Precision in Linear and Angular MeasurementsIn measuring angles and distances with a theodolite and a tape, one must maintain

consistency between the linear and angu-

lar measurements. As shown in Fig. 6.31,

let AB and AC be two lines. The angle

between the two lines is a and the length

of line AC is l. Let the angular error in

measuring the angle be da and the linear

error in measuring the length be n/l. The

true position of the required point is C, but

due to linear and angular errors, the posi-

tion shifts to C1. C2C1 is the linear error n.

aA B

C2

C1

C

da

Fig. 6.31 Precision in linear and angular measurements

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286 Fundamentals of Surveying and Levelling

CC2 = l tan da; C1C2 = n. Equating the two, tanda = l/n. Linear error is usually

expressed as 1 in Lvarious angles.

Table 6.13 Relationship between linear and angular error

Angular error 1/L Linear error Angular precision

10≤ 1/206,26 1 in 500 6¢ 50≤20≤ 1/103,13 1 in 3000 1¢ 8≤30≤ 1/6875 1 in 5000 41≤45≤ 1/4584 1 in 10,000 21≤60≤ 1/3438 1 in 25,000 8≤

this value is 1/20626 which is the corresponding linear error. If the linear error is

1 in 500 (1/500), tan–1

6.12 Upkeep and Maintenance of Theodolite

following points should be taken care of to maintain the instrument in good condi-

tion.

(a) The instrument should be properly kept in the box supplied with the

equipment. The clamp screws are loosened to enable movement and the in-

strument is then kept in the proper position in the box. The instrument is kept

in a tight position within the box by appropriate holding knobs or plates. The

instrument should not move around when it is carried in the box.

-

(c) It should be protected by a plastic cover or any other similar arrangement when

kept at a station while taping or locating details.

(d) Depending upon the extent of usage, the instrument should undergo permanent

adjustments frequently.

prevent it from falling off the tripod.

(f) Anti-moisture gel must be kept in the box to prevent damage to the optical

parts due to excessive humidity.

(g) The instrument should be dusted to remove dust and moisture before keep

ing it in the box after the day’s work.

If proper care is taken the theodolite can provide good service for a long time.

6.13 Modern Theodolites

We have discussed the vernier theodolite earlier. The vernier theodolite was a

very valuable instrument in earlier times. Most vernier-type instruments have a

least count of 20≤. They are still in use in many places. For geodetic work, vernier

instruments with large-diameter circles were used with a small least count in early

days. To improve this instrument and measure angles with greater precision, many

and electronic theodolites, which need less maintenance, were developed for ease

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Theodolite Surveying 287

of operation and greater precision. In these theodolites, permanent adjustments

remain for a longer period of time. Angles can be read to an accuracy of 1≤ or even

vary with the manufacturers but the essential principles remain the same.

6.13.1 Micro-optic TheodolitesMicro-optic theodolites can read angles to an accuracy of 10≤ or even less. The es-

sential principle is illustrated in Fig. 6.32. The special features of such theodolites

are as follows.

(a) Conventional metal circles are replaced by glass circles on which the gradua-

and sharper by this technique. Both the horizontal and vertical circles are made

of glass and generally graduated to 10¢.(b) Light passing through the circle at the point of the reading is taken through a

circle. In case night operation is required, the battery-operated light provided

in the instrument can be used.

of view. This is an advantage, as the readings of both the circles can be taken

at the same time. Some manufacturers make a switching arrangement so that

the horizontal or vertical circle reading can be seen along with the micrometer

reading.

(d) The optical micrometer is used to read fractions of the main scale division.

Depending upon the reading system, angles can be read up to 10¢ or less.

(e) The circles are generally graduated to 10¢ or 20¢ of the arc. The micrometer

can be read after coinciding the index with the nearest main scale division.

The fractions are then read from the micrometer scale, which is also seen in

(f) A small, separate reading telescope is provided besides the main telescope. It

eliminates the need to move while bisecting an object and taking the reading.

(g) In most instruments, diametrically opposite ends of the circle are brought

Reading telescopeMicrometer

Verticalcircle D

C

A BHorizontal circle

Fig. 6.32 Micro-optic theodolite

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288 Fundamentals of Surveying and Levelling

6.13.2 Electronic TheodolitesThe electronic theodolite (Fig. 6.33) is a recent development in the manufacture

of theodolites. The working principle remaining the same, the graduated circle is

made differently and the method of taking readings is also different. At the press

of a button, the angles are displayed and can be stored in memory. The following

are the characteristic features of electronic theodolites.

Movablephotodiode

Fixed photodiodeCoded circle

Laser digital theodolite

Fig. 6.33 Incremental encoding

Graduated circleThe graduated circle is made of glass as in an optical theodolite, but the graduations

are made in a special code to be read by photodiodes. The photodiodes convert the

readings into electrical signals that are processed by a microprocessor into angles

that are digitally displayed on an LCD or LED panel. The graduated circle can be

encoded in two ways—the incremental system and the absolute encoding system.

Incremental system In this system, two photodiodes are placed at diametrically

opposite positions over the glass circle. The circle is encoded by dividing it into a

zero of the circle. As the telescope is turned in azimuth, the other photodiode moves

along with the telescope. The circle does not move. The photodiode measures the

movement of the alidade from the code etched on the circle and the signal is sent to

a processor for converting the measurement into an angle for display and storage.

Absolute encoding system In this system, each position on the circle is uniquely

encoded. The circle is illuminated by an internal source and this light passes through

the circle onto an array of photodiodes, which convert the light into an electric signal.

This signal is processed by a microprocessor for display as an angle and storage.

CompensatorMost optical and electronic theodolites have a compensator system to ensure that

any deviation of the axis from the true horizontal does not affect the reading. Single

or dual-axis compensators are employed. Electronic theodolites are levelled as in

the case of vernier theodolites. The vertical axis may not exactly coincide with the

true vertical. To make the levelling process simple, compensators are provided.

The theodolite is only approximately levelled and any small tilt of the instru-

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Theodolite Surveying 289

ment is taken care of by the compensator. The compensator systems are shown in

Fig. 6.34.

Collimator

Photodetector

Level vial

Detection coil

Excitation coil

Level vial

Collimator

(a) Single-axis compensator (b) Dual-axis compensator

Fig. 6.34 Compensator systems

The single-axis compensator measures the tilt of the vertical axis and com-

pensates by adjusting the vertical angle. The dual-axis compensator measures the

tilt of the vertical axis as well as the trunnion axis. The horizontal angle is also

adjusted for any tilt in the trunnion axis. Single-axis compensator systems are of

liquid or magnetic type. Current is passed through the central coil known as the

excitation coil. This generates a current in the other two coils. The current values

will be equal if the instrument is level and different if there is any tilt in the vertical

axis. This difference is measured and converted into a tilt angle. The measured angle

is also compensated for this tilt by the processor before displaying the angle.

In dual-axis compensators, light from an internal source is passed onto a lens

for evenly distributing the light onto a liquid vial. A photodiode divided into four

sections receives this light. The light falling on the various sections of the pho-

todiode will be even if there is no tilt in the axes and uneven if there is any tilt in

the axes. The difference in light intensity is converted into tilt angles for both the

axes by the processor. The horizontal and vertical angles are also corrected for tilt

before displaying.

Display consoles may be provided either on one side or on both sides. Most

electronic theodolites have a zero set facility—changing the direction of the read-

ing from clockwise to counterclockwise, a display of angles, a display of grades,

etc. The angles are measured and displayed within a few seconds of pressing a

button. The batteries last for about 75 to 100 hours of operation. The data from the

theodolite can be transferred to a computer for further processing.

Digital theodolites

The instruments are light weight and are similar to electronic theodolites in con-

struction.

The instrument is set up over a station as in the case of normal theodolites. They

will have extendable tripod legs which can be adjusted for comfortable viewing.

The centring and levelling operations are done with a circular vial for coarse setting

one has to press only a measure button to get the readings of angles and distances.

Some models also have a laser pointer for easy alignment in critical cases and for

staking out operations. With the arrival of total stations, these theodolites have less

demand though they are cheaper compared to a total station.

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The following are typical features in a digital theodolite:

∑ Angle measurement – by absolute encoding glass circle;

Diameter – 71 mm

∑ Horizontal angle – 2 sides; vertical angle–one side;

∑objective lens – 45 mm

∑ Field of view – 1°30¢ Minimum focus distance – 1m

∑ Stadia values: Multiplying constant – 100; additive

constant – 0

∑ Laser pointer – coaxial with telescope; 633 nm class II

laser; Method – focusing for alignment and stake out operations

∑ Display on both sides; 7-segment LCD unit

∑ Display and reticle illuminated

∑∑

∑∑ Power supply – 4 AA size batteries; Operating times – Theodolite only – 140 hours

∑ Laser only – 80 hours; Theodolite + laser – 45 hours

∑ Weight – 4.2 kg

Optical plummet

Most instruments are provided with an opti-

cal or laser plummet for accurate centring.

The plummet system is shown in Fig. 6.35.

A small telescope is provided at the tribrach

level. The station point can be seen through

the telescope very accurately. The image of

the station point is brought to the view of the

centre of the instrument.

6.14 Adjustments of Theodolite

The fundamental lines of a theodolite are the

following.

1. Vertical axis

2. Horizontal axis or trunnion axis

3. Line of collimation or line of sight

4. Axis of plate level

5. Axis of altitude level

6. Axis of the striding level, if provided

The desired relationships among these lines are (see Fig. 6.36) the following:

(a) The axis of the plate level must lie in a plane perpendicular to the vertical axis.

Also see Plate 1

Station mark

Pickup lens

PrismEyepiece

Tribrach

Standards

Fig. 6.35 Optical plummet

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(b) The line of collimation must be perpendicular to the horizontal axis. The line of

collimation, the vertical axis, and the horizontal axis must intersect at a point.

(c) The horizontal axis must be perpendicular to the vertical axis.

(d) The axis of the altitude bubble must be parallel to the line of collimation.

(e) The vertical circle should read zero when the line of collimation is horizontal.

Horizontal or

trunnion axis

Line of collimator

Optical centre

Parallel

Altitude bubble axis

Vertical

circle

a

Plate bubble axis

Horizontal circle

Vertical axis

Cross hair

Fig. 6.36 Desired relationships between the fundamental lines of a theodolite

Each of these conditions (relationships) will result in a desirable performance of

the instrument as follows. Condition 1 ensures that the vertical axis is truly vertical

when the plate bubble is in the centre. Condition 2 ensures that when the telescope

is moved about the horizontal axis or transited, it moves in a vertical plane. Condi-

tion 3 ensures that the vertical angle measured will be free from the error caused by

axes not being parallel. Condition 4 ensures that the line of sight generates a vertical

plane when the telescope is transited. Condition 5 ensures that vertical angles will

be free from any index error due to the vernier.

6.15 Permanent Adjustments of Theodolite

The permanent adjustments of a theodolite are done in a prescribed order so that

one adjustment does not affect any other adjustment. The order in which the

adjustments are to be done is the following.

1. Plate level test to make the plate level at the centre when the vertical axis is

truly vertical.

2. Cross hair ring test to make the line of collimation coincide with the optical

axis and also to ensure that the line of collimation generates a vertical plane

when the telescope is transited.

3. Spire test to make the horizontal axis perpendicular to the vertical axis.

4. Collimation test to make the line of collimation perpendicular to the horizontal

axis.

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292 Fundamentals of Surveying and Levelling

5. Telescope bubble test to centre the telescope bubble when the line of sight is

horizontal.

6. Vertical vernier test to ensure that the vertical circle reads zero when the line

of sight is horizontal.

6.15.1 Plate Level Test

The plate level test is intended to make the vertical axis truly vertical. The steps are as

follows (see Fig. 6.37).

Test

level(s) and foot screws as in temporary adjustment.

2. Turn the instrument through 180°. Reverse the bubble end for end. If the bubble

does not traverse, adjustment is required.

be n.

B AA B(90° – )a

a

Vertical axisTrue

vertical

Bubble axis

Horizontal plate

(a) First position (b) Position after turning through 180°

aa

Truevertical Vertical axis

Fig. 6.37 Plate level test

Adjustment1. Shift the bubble towards the centre by n/2 divisions using the capstan-headed

screws at the end of the bubble tube. Bring the bubble to the centre of its run

using the foot screws.

2. Repeat the test and adjustment till the instrument is in adjustment.

Alternative method The altitude level on the telescope or on the T-frame is more

sensitive than the plate level. This is used for adjustment. The procedure is as follows.

Test1. Turn the telescope parallel to any two foot screws. Bring it to the centre of its

run using the two screws. Turn the telescope through 90° and bring the level

over the third foot screw. Bring it to the centre of its run. Repeat until the bubble

remains at the centre in these two positions.

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Theodolite Surveying 293

the number of divisions by which the bubble is off centre. Let the bubble be

off centre by n divisions.

Adjustment

1. Bring the bubble halfway (n/2 divisions) by means of the clip screw or the

vertical circle tangent screw. Bring the bubble to the centre of its run (correct

the remaining n/2 divisions) using the foot screws.

2. Bring the bubble over the third foot screw. Check whether the bubble traverses.

If not, adjust with the third footscrew.

3. Repeat the test and adjustment till the altitude bubble traverses in all positions.

4. The vertical axis has now been made truly vertical. If the plate level is off cen-

tre, bring it to the centre by means of the capstan-headed screws. The bubble

should now traverse in all positions.

6.15.2 Cross Hair Ring Adjustment

The cross hair ring adjustment test is done to make the line of collimation coincide

with the optical axis of the telescope. The vertical and horizontal hairs are adjusted

separately.

Adjustment of the horizontal hair For the adjustment of the horizontal hair, proceed as follows [see Fig. 6.38(a)].

Test1. Select two points P and Q about 100 m apart. Select a point R about 10 m from

P in line with P and Q on the line PQ.

2. Set up the theodolite at P and level the instrument carefully.

3. Keep a levelling staff each at R and Q with face left and take readings. Let the

readings be r1 and q1.

4. Transit the telescope and swing it through 180° to get the face right.

5. Set the horizontal hair to the earlier reading r1 obtained on the staff at R.

6. Again take the reading on the staff at Q. If the reading is the same as that ob-

tained earlier, q1, the instrument is in adjustment as far as the horizontal hair

is concerned. If not, let the reading be r2. Adjustment is required for the hair.

Second lineof sight

First lineof sight

P R

r1

q1

q2

Q(a) Horizontal cross hair test

Fig. 6.38 (Contd.)

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294 Fundamentals of Surveying and Levelling

a

q2

q1

(b) Alternative horizontal cross hair text

100 m100 m

PR

a

12

3

23

(c) Adjustment of vertical hair

Q

a

2a

Fig. 6.38 Cross hair ring test

Adjustment1. With the top and bottom screws on the diaphragm, bring the reading to the

mean reading (r1 + r2)/2.

2. Repeat the test till the instrument is in adjustment.

Alternative method As an alternative, the horizontal angle can be noted while

taking the reading. Proceed as follows [see Fig. 6.38(b)].

Test1. Select two points P and Q about 100 m apart from each other. Set up and level

the instrument at P.

reading of the vertical angle.

3. Transit the telescope and turn it through 180° to sight the staff again at Q with

the alternative face. Set the reading on the vertical circle to the same angle.

Read the staff. If the reading is the same as q1, the instrument is in adjustment.

If not, let the reading be q2.

Adjustment1. The adjustment is done using the diaphragm screws by setting the reading to

the mean of the two readings, (q1 + q2)/2.

2. Repeat the test and adjustment until the instrument is in adjustment.

Adjustment of the vertical hair Vertical adjustment is done to ensure that when the telescope is transited, it generates

a vertical plane. If the adjustment is not done, it will generate a cone with its axis

as the horizontal axis. We proceed as follows for this adjustment [see Fig. 6.38(c)].

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Test1. Select a point R such that about 100 m of level ground is available on either

side of R. Set up and level the instrument at R. This is to ensure that the test

is not affected by the difference in elevation between the points on either side.

Any further adjustment of the horizontal axis, even if it is not strictly horizontal,

will not affect the adjustment.

2. Select a point P about 100 m from R. Keep a ranging rod at R and bisect it

accurately with the telescope. Keep the upper and lower motions clamped.

3. Transit the telescope. Keep a ranging rod at about 100 m from R on the other

side of P in the line of sight. Place the ranging rod accurately in line with PR.

4. Transit the telescope again. Check whether P is in the line of sight. If P is sighted

accurately, the instrument is in adjustment. Otherwise, the vertical hair has to

be adjusted.

Adjustment1. Place a ranging rod in the line of sight beside P. Let this position be P¢. Measure PP¢.

to a point P≤ at one-fourth distance of PP¢ from P.

3. Repeat the test till the instrument is in perfect adjustment.

It must be noted that the instrument is transited twice and the apparent error is four

times the actual error.

6.15.3 Spire Test In the spire test the horizontal axis is

made perpendicular to the vertical axis

when the instrument is levelled. This,

together with the adjustment of the

vertical hair, ensures that the line of

collimation generates a vertical plane.

Proceed with the test and adjustment as

follows (see Fig. 6.39).

Test1. Set up the instrument at 10–20 m from a building or any structure that has a clearly

2. Set up and level the instrument at this point. With face left, sight the high point.

Clamp the motions and depress the telescope. Mark a point on the ground or

on the wall. It may be better to use a levelling staff placed horizontally below

the high point. In such a case, note the reading on the staff.

3. Transit the telescope, turn it through 180°, and again sight the point on the top.

it is the same as the previous one, the instrument is in adjustment; otherwise,

adjustment is required.

Adjustment1. If points are marked, measure the distance between the points on the wall or

ground. Find the point midway between the two marks. If a staff is used, take

Second mark

Third mark

Correct

Fig. 6.39 Spire testPreview

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296 Fundamentals of Surveying and Levelling

the mean of the two readings to get the point midway between the two readings.

Sight this point with the telescope. Clamp both motions.

2. Raise the telescope and check whether the high point is sighted. It will not

bisect the point.

3. With the horizontal axis adjustment screw on the standards, adjust the horizontal

axis so that the point is sighted.

4. Repeat the test and adjustment till the instrument is in adjustment.

6.15.4 Collimation Test The collimation test ensures that the line of sight coincides with the optical axis

of the telescope. The test procedure is the same as the two-peg test described for

a level.

Alternative method The procedure is the same as in the two-peg test except

for the following.

1. With the vertical circle clamped, bring the bubble to the centre of its run using

the clip screw.

2. In adjusting the reading on Q to the correct reading, use the clip screw. This

eliminates any index error in the vertical circle and vernier.

6.15.5 Altitude Bubble TestThe objective of the altitude bubble test is to ensure that the vertical circle reads

zero when the telescope bubble is centred. If the telescope bubble is in the centre,

but the vertical circle does not read zero, then there is an index error. Corrections

can be applied to the vertical angles measured. However, the test and adjustment

are done to eliminate the index error so that confusion of signs in applying the cor-

rections is avoided. Proceed as follows.

plate bubble.

2. Bring the altitude bubble to the centre of its run with the help of the vertical

circle tangent screw.

3. Read the vertical circle. If it reads zero, the instrument is in adjustment.

4. If the vertical circle does not read zero, note down the reading. This is the

index error, which should be added to or subtracted from the vertical angles

measured.

5. To make the vertical circle read zero, the screws attaching the vernier arc

to the standards are loosened and adjusted to make the vertical circle read

zero.

6.16 Importance of Adjustments

The error caused by many of the required conditions of the fundamental lines not

being met can be eliminated by face-left and face-right observations.

The plate level adjustment is very important for all the operations of the the-

odolite. This adjustment should be done frequently. If the vertical axis is not truly

vertical, the error cannot be eliminated by face-left and face-right observations.

The plate level adjustment ensures that the horizontal circle and trunnion axis are

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Theodolite Surveying 297

truly horizontal. The measurement of horizontal angles is not affected except when

using the method of repetition.

The second adjustment is that of cross hairs. The adjustment of the horizontal

hair is important for measuring vertical angles and also when the theodolite is used

as a level. If the horizontal hair does not lie on the optical axis of the telescope, its

position will change during focusing. This, however, does not affect the measure-

ment of horizontal angles. The adjustment of the vertical hair is important in many

operations such as prolonging a line by transiting, measuring horizontal angles,

and measuring with the telescope in inclined positions. The errors caused can be

eliminated by face-left and face-right observations.

The third adjustment to make the horizontal axis perpendicular to the vertical

axis becomes important in all operations involving the motion of the telescope in a

vertical plane. The telescope should generate a vertical plane during such motion,

which is ensured by this adjustment.

The fourth adjustment, of the altitude level, is important for accurate measure-

ment of vertical angles and for using the theodolite as a level.

The index error adjustment of the vertical circle and vernier is done for conve-

nience only. It is possible to apply corrections to the measured angles after noting

down the index error.

Points to Remember

∑ Theodolite is used to measure horizontal and vertical angles. A theodolite consists of a base lev-

elling head for levelling the instrument and can be screwed on to a tripod stand. The horizontal

circle is graduated from 0 to 360 degrees and has a vernier to read angles up to 20”. Magnifying

glasses are provided to read the two verniers provided 180 degrees apart.

∑ The A-frame on top of the levelling head carries the telescope and the vertical circle graduated

from 0 to 90 degrees in the four quadrants. There is an altitude bubble for levelling which is

more sensitive than the horizontal bubble on the levelling head. The telescope can be transitted

or rotated in a vertical plane and can be moved horizontally.There are locking screws and tangent

attached for observing bearing of a line.

∑ Temporary adjustments of the theodolite to be performed at each station where it is set up are

centering and levelling. Centering is done using a plumb bob attached to the lower hook on the

levelling head by adjusting the tripod legs. Levelling is done by the spirit level on the levelling

head and bringing the bubble to the centre in two perpendicular directions. Before taking observa-

tions both the eye piece, for a clear image of the cross hairs, and objective lens, for a clear image

of the object, have to be adjusted.

∑ Permanent adjustments are done to maintain the desired relationships between the fundamental

lines of a theodolite. The adjustments are done in a particular order to maintain the earlier adjust-

ments.

∑ Face left and face right observations using a theodolite are taken when the vertical circle is

to the left or right of the telescope. The observation face is made to change by transitting the

telescope and panning the telescope to the object.

∑ Horizontal angles are measured using the repetition method by measuring the angle a number of

times from different parts the circle. Bothe face left and face right observations are made. Another

method is the reiteration method where a number of angles are measured at a point closing the

horizon. For accurate work, a number of sets of observations are taken.

∑ Vertical angles, either angles of elevation, are made by tilting the telescope and bisecting the

point with the horizontal cross hair. The altitude bubble can be used to level the instrument for

this measurement.

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298 Fundamentals of Surveying and Levelling

∑ Many other operations like prolonging a line and ranging of a line can be performed using the

theodolite.

∑ Traversing is the main operation done using a theodolite. Traversing is done by measuring interior

∑ In a closed traverse, interior angles must add up to (2n – 4) right angles; exterior angles must

∑ In adjusting a traverse, any difference in angles is equally distributed among the angles.

∑ The fundamental lines of a theodolite are: vertical axis formed by the inner and outer axes of

the leveling head, trunnion axis on which telescope is supported, bubble tube axis of the plate

level and altitude level and the line of collimation of the telescope.

∑ Modern theodolites

angles through the main or an additional reading telescope. The view of the circles are brought

glass circles read by photodiodes. The display panel shows the angles. For centering, an optical

plummet is generally used.

Exercises

Multiple-Choice Questions1. A theodolite is often designated by

(a) diameter of the objective lens (c) diameter of the eye piece lens.

(b) diameter of the horizontal circle (d) length of the telescope

2. To change the reading on the circle while measuring an angle,

(a) upper clamp is tightened and lower clamp is loosened

(b) upper clamp is loosened and lower clamp is tightened

(c) both upper and lower clamps are loosened

(d) both upper and lower clamps are tightened

3. The horizontal circle in a theodolite is graduated in

(a) the quadrantal system from 0 to 90 in the four quadrants

(b) the whole circle system from 0 to 360

(c) the semi-circle system from 0 to 180 in the right and left halves

(d) a way similar to that in a prismatic compass

4. The vertical circle of a theodolite is generally graduated in

(a) the quadrantal system from 0 to 90 in the four quadrants

(b) the whole circle system from 0 to 360

(c) the semi-circle system from 0 to 180 in the right and left halves

(d) a way similar to that in a prismatic compass

5. When you transit the telescope, you rotate the telescope about

(a) the vertical axis (c) the optical axis of the telescope

(b) the trunnion axis (d) the line of collimation

6. A face-left or normal position of observation means

(a) the vertical circle of the instrument is on the left of the observer

(b) the vertical circle is on the right side of the observer

(c) the station being observed is to the left of the observer

(d) the left side vernier is read for the angle value

7. Focusing the objective lens is an operation done

(a) to get the cross hairs to clear view

(b) every time to focus the object being sighted

(c) for reading the vernier clearly

(d) for reading vertical angles

8. In the method of repetition for measuring horizontal angles, to rotate the instrument without

changing the reading,

(a) lower clamp screw is tightened and upper clamp is loosened

(b) lower clamp screw is loosened and upper clamp screw is tightened

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Theodolite Surveying 299

(c) any one of the clamp screw is loosened

(d) both the clamp screws are loosened

9. In the method of reiteration of measuring horizontal angles,

(a) the same angle is measured three times

(b) the same angle is measured by face-left and face-right observations

(c) the angle is measured and the instrument turned to close the horizon

(d) the angle is measured three times each using face-left and face-right observations

10. The inner axis of a theodolite is

(a) the plate bubble axis on the horizontal circle

(b) the axis passing through the centre of the vertical circle

(c) the trunnion axis

(d) the centre of the spindle carrying the vernier circle

11. The outer axis of a theodolite is

(a) the axis of the altitude level

(b) the trunnion axis

(c) the axis passing through the centre of the horizontal graduated circle

(d) the line of collimation of the theodolite

12. The index frame or vernier frame is

(a) the A-frame attached to the telescope

(b) the verner circle of the horizontal circle

(c) the T-shaped frame carrying the vernier of the vertical circle

(d) the base of the theodolite having the leveling head

13. A micro-optic theodolite has horizontal and vertical circles made of

(a) steel (c) aluminium

(b) glass (d) horizontal and vertical circles made of light plastic

14. In electronic theodolites, the horizontal and vertical circles are made of

(a) glass having specially coded graduations read by photo diodes

(b) special metal read electronically

(c) steel plates with specially coded graduations

(d) plastic with coded graduations

15. Optical plummet is used

(a) in optic theodolites for measuring angles

(b) in electronic theodolites for compensating any tilt in the axis

(c) for accurate centrring of the theodolite over a station

(d) for accurate levlling of the theodolite

16. The horizontal axis of the theodolite is the line

(a) tangential to the plate bubble

(b) tangential to the altitude level

(c) perpendicular to and passing through the centre of the vertical circle

(d) the line joining the intersection of cross hairs and centre of objective lens

17. Plate level test of a theodolite is done to ensure that

(a) horizontal axis is truly horizontal

. (b) the vertical axis is truly vertical

(c) the plate bubble axis is horizontal

(d) the horizontal circle is truly horizontal

18. The spire test for permanent adjustment of a theodolite is done to ensure that

(a) the line of sight is perpendicular to the vertical axis

(b) the two axes, of plate level and altitude level, are parallel

(c) the vertical circle lies in a vertical plane

(d) the horizontal axis is perpendicular to the vertical axis

19. The error that cannot be eliminated by face left and face right observations is when the

(a) vertical axis is not truly vertical

(b) line of collimation and trunnion axes are not perpendicular

(c) axes of the bubble tubes are not parallel

(d) line of collimation and axis of plate level are not parallel

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300 Fundamentals of Surveying and Levelling

Review Questions1. Draw a neat sketch of a vernier theodolite. Describe its main parts and their functions.

lines and the effects if such relationships are not maintained.

3. With a neat sketch, explain how the main scale and vernier scale are graduated to obtain a least

count of 20≤.

4. Explain the temporary adjustments of a theodolite.

5. Explain the steps involved in measuring horizontal angle with a theodolite.

6. Explain the repetition method to measure horizontal angles and how readings are recorded.

7. Explain the procedure for the reiteration method of measuring horizontal angles.

8. Explain the step-by-step procedure to measure the vertical angle between two points.

10. Explain the procedure to measure the bearing of a line with a theodolite.

12. Describe a method to range a line between two points when the stations are not intervisible from

an intermediate point.

13. Explain the different methods to prolong a line.

be taken to eliminate them.

15. List the possible personal errors in theodolite work and the precautions that should be taken to

eliminate them.

16. Describe how consistency in linear and angular measurements can be achieved.

Problems1. The following included angles were measured in a closed traverse. If the bearing of the line AB is

70° 40¢ 30≤ Station A B C D E

Angle 78° 42¢ 30≤ 104° 15¢ 20≤ 92° 44¢ 40≤ 112° 36¢ 10≤ 131° 41¢ 20≤2. The bearings of lines measured in a theodolite survey are given below. Find the included angles

of the traverse.

Line PQ QR RS

Bearing 160° 33¢ 20≤ 58° 47¢ 40≤ 320° 46¢ 30≤

Bearing 280° 27¢ 30≤ 225° 52¢ 40≤ 114° 58¢ 20≤

Station A B C D E

Angle 102° 30¢ 40≤ 124° 10¢ 40≤ 96° 08¢ 20≤ 131° 00¢ 40≤ 86° 09¢ 40≤

traverse.

Station A B C

¢ 20≤ 70° 18¢ 40≤ 88° 20¢ 20≤ Station D E

¢ 10≤ 82° 31¢ 30≤¢ 20≤,

Station P Q R

¢ 40≤ 68° 30¢ 20≤ 81° 18¢ 30≤ Station S T

¢ 10≤ 94° 00¢ 20≤6. The exterior angles of a traverse were measured during the survey of an area. If the bearing of

the line PQ is 45° 32¢ 40≤ Station P Q R

Ext. angle 202° 36¢ 20≤ 250° 18¢ 40≤ 268° 20¢ 20≤ Station S T

Ext. angle 276° 13¢ 10≤ 262° 31¢ 30≤

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