1. Fundamentals of Relative Directions

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Lecture-1

Fundamentals of Relative Direction

Introduction

In order to get the location of a point in surveying on, above or below the surface of the earth, itis necessary to know its distances from a point / station along a line whose direction is known.

Fundamental Concepts

The direction of a line is defined by a horizontal angle with respect to a reference line.Depending on the type of reference, the direction of the line is termed as relative or absolute. In

relative direction, the reference line does not remain fixed over time.

Meridian

The reference line with respect to which horizontal angle of survey line are measured is known

as meridian. There are different types of meridians based on the type of reference.

True Meridian Magnetic Meridian

Grid Meridian

Arbitrary Meridian

True Meridian

The true meridian passing through a station on surface of the earth is the (imaginary) line ofintersection of a (imaginary) plane passing through the geographical North and South poles of

the earth with its actual surface. The direction of true meridian at any station is constant andhence, direction of a line with reference to this remains same over time. At any station, it can
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be determined through Astronomical survey. For any engineering works of importance, thedirection of at least one line is determined with true meridian as reference.

Magnetic Meridian

The magnetic meridian at a station on surface of the earth is the (imaginary) line of intersection

of a (imaginary) plane passing through the magnetic North and South poles of the earth with itsactual surface. The direction of a freely suspended well balanced magnetic needle provides the

magnetic meridian at a station. As the magnetic poles of the earth changes with time and so themagnetic meridian at any station. Thus, the direction of a line with reference to magnetic

meridian varies with time. Magnetic meridian is employed as a line of reference for roughsurveys.

Grid Meridians

To eliminate the effect ofconvergence of meridian,a meridian through a station is adopted as areference meridian and all other meridians are considered parallel to the reference meridian.

Together, all of these are known as grid meridians and the reference meridian is known ascentral meridian.

Convergence of meridian :Meridians on the surface of the earth converge towards each other

as the distance from the equator towards either of the poles increases.

Arbitrary Meridian

Any convenient direction from a survey station to some well defined permanent object is

known as arbitrary meridian. This is used for small area survey or to determine the relativedirections of small traverse.

Relative Direction

The direction of a line is expressed in different ways depending upon the type of survey. It isdepicted by bearing, included angle, deflection angle etc. These quantities may be observed

directly in the field or can be obtained indirectly by computation.

Bearing

The horizontal angle measured in clockwise or anticlockwise direction between the meridian

and the survey line is termed as bearing. Different types of bearings are defined based ondifferent criteria.

Based on meridian

Azimuth

Magnetic Bearing

Grid Bearing

Arbitrary Bearing
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Based on direction

Fore Bearing

Back Bearing

Based on designation

Whole circle bearing

Quadrantile bearing (or Reduced bearing)

Azimuth or True Bearing

The azimuth or true bearing of a line is its horizontal angle from the North direction of the true

meridian measured clockwise.

In Figure 19.2, azimuth of a line OA is given by NOA (= 52), measured from the North(Geographical) and that of line OB is NOB (= 208)

Magnetic Bearing

The horizontal angle which a line makes with the magnetic meridian measured from MagneticNorth line is called magnetic bearing. It varies with time. Magnetic meridian of a line can be
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measured in the field by using prismatic compass (Figure 19.3)

Grid Bearing

The horizontal angle between a line and a grid meridian is called grid bearing.

Arbitrary Bearing

The horizontal angle of a line measured with respect to an arbitrary meridian is called arbitrarybearing.

Based on the Direction

Any straight line has two diametrically opposite directions. The direction in which a survey work

proceeds is known as forward direction and the opposite direction is known as backwarddirection. Thus, there are two types of bearing depending upon the direction of the line for which

it is being is determined.

1. Fore Bearing

2.

Back Bearing

Fore Bearing

The bearing of a line measured in the forwrard direction (i.e., along the progress of survey) is

known as fore bearing.
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InFigure 19.4,fore bearing of the line AB is given by NOB.

Fore bearing = Back bearing 180

Back Bearing

The bearing of a line measured in the backward direction (i.e., opposite to the direction ofprogress of survey) is known as back bearing.

In Figure 19.5, the back bearing of the line AB is NOA (= 223). NOA is also calledbearing of the line BA. Thus,

Back Bearing = Fore Bearing 180
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Designation of Bearing

The angle representing bearing is designated depending on the measurement of the angle eitherin clockwise and anti-clockwise direction measured either from the North or from the South limbwhichever provides minimum angle.

Whole circle bearing Quadrantal bearing (or Reduced bearing)

Whole Circle Bearing

The whole circle bearing (W.C.B) of a line is the horizontal angle measured clockwise from theNorth limb of the meridian. It varies from 0 to 360.

InFigure 19.6,The whole circle bearing (W.C.B) of the line OA is 52 and that of line OB is208
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Quadrantal Bearing (or Reduced Bearing)

The quadrantal bearing (Q.B.) also known as reduced bearing (RB) of a line is defined by theacute angle which the line makes with the meridian. Thus, it depends on the quadrant in which

the line presents. It is measured in clockwise or anti-clockwise direction either from the North orfrom the South limb of the meridian whichever is nearer and thus provides minimum angle.

Figure 19.7represents different types quadrantal bearings of lines. The bearings of OP, OQ, ORand OS are respectively, N 40E, S49 E, S72 W, and N31 W. In all cases, values of bearing of

angles lie between 0 and 90.

Thus, reduced bearing of a line is designated by the direction from which it is measured (i.e.,

either N for North or S for South) followed by the value of the angle at the end, the direction to

which it is measured (i.e., either E for East or W for West)
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Conversion of reduced bearing (RB) to whole circle bearing (WCB) and vice versa

The RB of any line can be converted to its equivalent RB as shown in Table 19.1.

Table 19.1 Conversion of RB to WCB (Figure 19.8)

Quadrant Line QB or RB WCB

I OA Nq1 E q1

II OB Sq2 E 180 - q1

III OC Sq3 W 180 + q3

IV OD Nq4 W. 360 - q4

Similarly the WCB of a line can be converted to its equivalent RB as shown in Table 19.2.

Table 19.2 Conversion of WCB to RB (Figure 19.9)
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Quadrant Line WCB RB

I OA 0 q190 Nq1 E

II OB 90 q2180 S (180 -q2) E

III OC 180 q3270 S (q3 -180) W

IV OD 270 q4360 N (360 -q4) W

Interior Angle

In any closed polygon, the direction of any side can be depicted by the angle it makes with its

adjacent side. This angle measured clockwise or anti-clockwise is known as interior angle.Figure 19.10 shows interior angles at the stations A, B, C, D and E of a closed polygon. The

direction of the side AB can be obtained either of the included angles at vertices such as A, Betc.


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Deflection Angle

The angle between a line and the prolongation of the preceding line is called a deflection angle.Deflection angle can also be used to note the direction of a line. It is recorded as right or left

depending on whether the line to which measurement is taken lies to the right (clockwise) or left(counterclockwise) of the prolongation of the preceding line. Figure 19.11 shows deflection

angles. In this, the line BC is having a right deflection angle CD does have left deflestion angle.Deflection angles may have values between 0 and 180, but usually they are not employed for

angles greater than 90. In any closed polygon the algebraic sum of the deflection angles(considering right deflections as of sign opposite to left deflections) is 360


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Angles to Right

Angles are also depicted by clockwise measurement from the preceding to the following line, as

shown inFigure 19.12.Such angles are called angles to right.

Declination

The horizontal angle which a magnetic meridian makes with the true or astronomic meridian is

called the magnetic declination, or magnetic variation. If the north end of the compass needledeflects right of the true meridian, the declination is said to be east (Figure 19.13(a)) and if it

deflects to the left of the true meridian, the declination is said to be west (Figure 19.13(b)).Declination varies from place to place and at any station time to time
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Figure 19.13 Magnetic Declination

Correction for Declination

When magnetic directions are used to obtain estimates for azimuth or an old survey is required tobe retraced, it is necessary to reduce the magnetic directions to true bearings or azimuths by

making use of declination for necessary corrections.

Examples

Ex19-1The magnetic bearing of a line was recorded as N3400'E on an old survey plan dated

August 15, 1947. It is required to re-establish this direction on the site in 1997. In 1970 isogonicchart shows a declination 9 W for the area, with an annual change of l' eastward. Determine the

(i) Azimuth of the line;

(ii) magnetic bearing that must be used to relocate the direction of the line in the field.


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Solution :

Declination in 1970 = 9 W

change in (1970 - 1947) = 23 yrs (23 x 1') = 23' Eastwards

Therefore

Declination in 1947 was 9 + 23' = 9 23' W

Magnetic Bearing of the line on 1947 = N 34 E

(i) True bearing of the line = 34 - 9 23' = 24 37'

= N 24 37' E

Declination in 1997 = (9 - 27 x 1' ) W = 8 33' W

Therefore Magnetic bearing in 1997 = 24 37' + 8 33' W

= N 33 10' E


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Local Attraction

In presence of magnetic materials, the magnetic needle deviates from the magnetic meridian

and thus provides wrong direction of a line. The deviation arising from such local sources iscalled local attraction. If the fore bearing and back bearing of a line does not differ by 180,

then there is a possibility of local attraction during the observation of the line. Otherwise, ifthe sum of the interior angles of a closed traverse does not provide (2n - 4) right angles [where

n is the number of sides in the traverse], then there is a possibility of local attraction duringthe observation of the traverse.

Determination of Angles/ Directions

Direction and thus angles can be determined by means of a tape, plane-table, alidade, sextant,

or compass etc but normally these are measured with a transit theodolite. In this module,measurement of angles using theodolite has been considered

Ex19-2 Following are the observed magnetic bearings of the traverse legs:

Line AB BC CD DA

FB (RB) S 59 30' E N 78 15' E N 59 30' W S 30 15' W

BB (WCB) 300 30' 256 00' 125 15' 12 45'

At what stations local attraction is suspected? Determine the correct bearings of the traverse

legs and also calculate the included angles.

Solution :

Figure Ex19-2


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The FB of the lines are given in reduced bearing. Their equivalent WCB are

LineFB

RBRB WCB

AB S 59 30' E 120 30' W 300 30'

BC N 78 15' E 78 15' 256 00'

CD N 59 30' W 300 30' 125 15'

DA S 30 15' W 207 45' 27 45'

The FB and BB of any line differs exactly by 180, if the stations are free from local attraction.

In the given observation, the FB and BB of the line AB differs by 180 and thus stations A and

B are free from local attraction. The bearing of the lines observed at stations A and B may beconsidered to be correct.

Given, FB of BC = 78 15'

Therefore Correct, BB of BC = 78 15' + 180 = 258 15'

But, observed BB of BC = 256 00'

Therefore Error at C = 258 15' - 256 00' = - 215'

Correction at C = 2 15'

Observed FB of CD = 300 30'

Correction at C = + 2 15'

Therefore corrected FB of CD = 302 45' - 180 = 122 45'

and corrected BB of CD = 125 15'

Error at D = + 2 30'

or Correction at D = - 2 30'

observed FB of DA = 210 15'

Therefore corrected FB of DA = 210 15' - 2 30' = 207 45'

and corrected BB of DA = 207 45' - 180 = 27 45' (Checked)

Line Corrected FB Corrected RB

AB 120 30' 300 30'


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BC 78 15' 258 15'

CD 302 45' 122 45'

DA 207 45' 27 45'

Included angle

As the traverse is running anti-clockwise the included angle will be the interior angles.

Angle at A = F.B. of AB - B.B. of DA = 120 30' -27 45' = 92 45'

B = F.B of BC - B.B of AB = 78 15' - 300 30' = - 222 15' + 360 = 137 45'

C = F.B of CD - B.B of BC = 300 30' - 256 00' = 44 30'

D = F.B of DA - B.B of CD = 210 15' - 125 15' = 85 00'

Calculation of Bearing (2nd method)

Bearing of the line AB = 120 30' (correct)

B = 137 45'

-----------------------

Bearing of the line BC = 258 15' - 180 = 78 15' (since traverse is anti-clockwise)

C = + 44 30'

--------------------------

122 45'

Bearing of the line CD = 122 45' + 180 = 302 45'

D = + 85 00'

---------------------------

387 45'

Bearing of CD = 387 45' - 180 = 207 45'

A = + 92 45'

------------------------


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300 30'

Bearing of AB = 300 30' - 180 = 120 30' (checked)

Line FB BB Angle

AB 120 30' 300 30' A = 92 45'

BC 78 15' 258 15' B =137 45'

CD 302 45' 122 45' C = 44 30'

DA 207 45' 27 45' D = 85

[Note : In any traverse, running anti-clockwise, included angle at any stations = F.B. of the

forward line - B.B. of the backward line].

Exercise 19

Ex.19-1Tabulate the differences between different types of meridians along with differencesin their utilities.

Ex.19-2 What is magnetic declination ?

Ex.19-3 The magnetic bearing of a line was found to be N 60 30' W in 1992, when thedeclination was 5 10' E. find its present magnetic bearing, if declination is 3 W.

Ex.19-4The bearing taken for two lines are as follows:

Line Fore Bearing Back Bearing

AB S 37 30' E 322 30' (WCB)BC 223 15' (WCB) N 44 15' E

Compute the interior angle at B.

Ex.19-5 Following are the observed magnetic bearings of the traverse legs:

Line PQ QR RS SP

FB 74 20' 107 20' 224 50' 200 15'

BB 256 00' 286 20' 44 50' 126 00'

At what stations local attraction is suspected? Determine the correct bearings of the traverse

legs and also calculate the included angles.

Answers

For Exercise 19


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Ex.19-1N 52 20'

Ex.19-2 99 15'

Ex.19-5 At R and S;

Angles 106 45', 123 45', 64 30', 65 00'

1. Fundamentals of Relative Directions

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