Bearing of a line is the horizontal angle which a line makes with some reference direction also known as meridian.

The reference direction may be any of the following:

1. A true meridian

2. A magnetic meridian

3. An assumed meridian

1. True Meridian:

True meridian of a place is a direction indicat­ed by an imaginary circle passing round the earth through that place and the two (north and south) geographical poles. The horizontal angle be­tween a line and the true meridian is called true bearing of the line. It is also known as azimuth.

2. Magnetic Meridian:

The direction indicated by a freely sus­pended and properly balanced magnetic needle unaffected by local attrac­tive forces is called the magnetic meridian. The horizontal angle which a line makes with this meridian is called magnetic bearing or simply bearing of the line.

3. An Assumed or Arbitrary Meridian:

For small surveys, any convenient direction may be taken as a meridian. It is usually the direc­tion of the first line of survey of the direction from a station to some well- defined permanent object. Such a meridian also has the advantage of being invariable and its direction can be recovered whenever required if the stations defining it are permanently marked or fixed by the measure­ments from permanent objects.

The horizontal angle between a line and this meridian is called assumed or arbitrary bearing of the line.

#### Designation of Bearings:

The bearings are expressed in the following two ways:

A. Whole circle bearings

A. Whole Circle Bearings:

The horizontal angle which a line makes with the north direction of the meridian measured in the clockwise direc­tion and can value upto 360° i.e. the whole circle, is known as whole circle bearing (W.C.B.) of the line.

The prismatic compass measures the bearings of lines in the whole circle system.

The horizontal angle which a line makes with the north or south direction of the meridian whichever is nearer the line measured in the clockwise or counter clockwise direction towards east or west and can value upto 90° i.e. one quadrant of a circle is known as quadrantal bearing of the line.

The surveyor’s compass measures the bearings of lines in the quadran­tal system.

Reduced Bearings:

The whole circle bearing when exceeds 90° may be converted or reduced into the corresponding bearing in the quadrantal system, which has the same numerical values of the trigonometrical func­tions. The bearing thus obtained is known as the reduced bearing (R.B.).

The following table may be referred to convert the whole circle bearings to the reduced bearings:

Fore and Back Bearings:

Every line has two bearings one, observed at each of the line. The bearing of a line taken in the progress of the survey or in the forward direction is the fore or forward bearing (F B.) of the line; while its bearing taken in the reverse or opposite direction is known as reverse or back bearing (B.B.)

The bearing of a line AB (Fig. 5.6) expressed in the direction A to B, (observed at A by sighting towards B) F.B. of AB. The bearing of AB / when recorded in the opposite / direction i.e. from B to A is B.B. of AB or F.B. of BA.

In the whole circle system, the fore and back bearing of a line differ exactly by 180°.

...B.B. of a line = F.B ± 180° …………………………………….. (Eqn 5.1)

Use Plus sign if the given F.B. is less than 180°; and minus sign if it exceeds 180°.

In the quadrantal system, fore and back bearings are numerically equal but with opposite cardinal points. Back bearings of a line may, therefore, be obtained by simply substituting N for S, or S for N; and E for W or W for E in its fore bearing. Supposing F.B. of a line is N 30°E, then its B.B. is equal to S 30°W.

Examples on Bearings:

Example 1:

Convert the following whole circle bearings to re­duced bearings.

(i) 65° – 30°; (ii) 140° – 20′ ; (iii) 255° – 10′ ; (iv) 336° – 40′.

Solution:

Applying the rules given in the table 5.1.

(i) W.C.B. = 65° – 30′; which is less than 90°

...R.B. = N (the same as W.C.B.) E = N 65° – 30′ E (Ans.)

(ii) W.C.B. = 140° – 20’; which is between 90° and 180°

...R.B. = S (180° – 140° 20′) E = S 39° 40′ E (Ans.)

(iii) W.C.B. = 255° – 10′: which is between 180° and 270°

...R.B. = S (255°10′ – 180°) W = S75° – 10′ W (Ans.)

(iv) W.C.B. = 336° – 40′; which is between 270° and 360°

...R.B. = N (360° – 336″40′) W = N 23° – 20′ W (Ans.)

Example 2:

Convert the following reduced bearings to the whole circle bearings:

(i) N 56° – 30′ E ; (ii) S 32° – 15’E ; (iii) S 85° – 45’W ; (iv) N 15°- 10′ W.

Solution:

(i) R.B. = N 56° – 30 E &, which is in the NE quadrant,

...W.C.B. = the same as R.B. = 56° – 30′ (Ans.)

(ii) R.B. = S 32° – 15′ E, which is in the SE quadrant,

...W.C.B. = 180° – 32° 15′ = 147° – 45′ (Ans.)

(iii) R.B. = S 85° – 45′ W, which is in the SW quadrant,

...W.C.B. = 180° + 85° 45′ = 265° – 45′ (Ans.)

(iv) R.B. = N 15° – 10′ W, which is in the NW quadrant,

...W.C.B. = 360° – 15° 10′ = 344° – 50′ (Ans.)

Example 3:

Find back bearings of the following observed fore hearings of lines AB, 63° – 30′; BC, 112° – 45′ ; CD, 203° – 45′ ; DE, 320′ – 30′.

Solution:

From the Eqn 5.1., B.B. = F.B ± 180°

F.B. of AB = 63°- 30′, which is less than 180°

... B.B. of AB = 63° 30′ + 180° = 243° – 30′ (Ans.)

F.B. of BC =112° – 45′, which is less than 180°

... B.B. of BC = 112°45′ + 180° = 292° – 45′ (Ans.)

F.B. of CD = 203° – 45′, which is less than 180°

... B.B. of CD = 203°-45′-180° =23°-45′ (Ans.)

F.B. of DE = 320° – 30′, which is more than 180°

... B.B. of DE = 320° 30′ – 180° = 140 – 30′ (Ans.)