Whole Circle Bearing (WCB)and Quadrantal Bearing System (QB)

Bearings in surveying can be designated by two methods as Whole Circle Bearing System (WCB) or Azimuthal System and Quadrantal Bearing System (QB) or Reduced Bearing System.

Before proceeding with this video, it is important to have a clear understanding of the terms "meridians," "true north," and "magnetic north." We recommend watching our previous videos or articles on magnetic meridians and bearings to fully grasp the concepts.

Watch Video: Whole Circle Bearing (WCB) & Reduced Bearing  

Table of Contents

1. Whole Circle Bearing System (WCB) or Azimuthal System
2. Reduced Bearing or Quadrantal Bearing (QB) System
3. Conversion Between WCB and QB system
4. Example Problems

1. Whole Circle Bearing (WCB) or Azimuthal System

A whole circle-bearing system involves measuring the angles from the true north or the magnetic north in the clockwise direction. The bearing or angle measured from the WCB system is called the azimuth.

As shown in the figure, we need to determine the azimuth of the line OA and OB. The azimuth of the line is measured from the north, giving a value of θ1 = 60 degrees and that of OB is θ2=220 degrees. 

 

Whole Circle Bearing (WCB) System

Hence WCB value varies between 0 and 360 degrees and in most cases, the reference meridian is the north. 

A prismatic compass that is used for compass surveying gives horizontal angles based on the WCB system.

In plane surveying, the azimuths are measured from the North. The National Geodetic Survey of the National Oceanic and Atmospheric Administration (NOAA) (formerly the United States Coast and Geodetic Survey) always uses the south as the zero direction.

2. Quadrantal Bearing (QB) or Reduced Bearing (RB) System

The quadrantal bearing system measures angles eastwards or westwards from the north or south direction whichever is nearer. For this, the QB system divides bearings into quadrants: North, East, South, and West. The obtained QB is called a reduced bearing and is expressed by mentioning the meridian, angle, and quadrant.

Quadrantal bearing system (QB)

In the figure, there are four quadrants ( I, II, III, IV). The line OA is present in the first quadrant, and the nearest meridian to this line is North. Hence, the bearing is measured from ‘N’ north in the clockwise direction i.e. eastwards, which is “ α = 30 degrees “. This is represented as “N α E” Similarly, the line OB is in the second quadrant, whose nearest meridian is “South”. Then the angle is measured from south to line OB in the anti-clockwise direction, eastwards i.e. β=50 degree, which is represented as “S β E”.

Hence, the reduced bearing values vary between 0 and 90 degrees. Also, both north and south meridians are used as reference meridians, and the directions of measuring can be the clockwise or anti-clockwise direction.

To summarise, as shown in the figure below, the azimuth of the line OA as per WCB is 150 degrees. But the RB value as per QB system is 30 degrees designated as S 30 E.   

Conversions of Bearings from One System to the Other

The bearings of a line can be available either in WCB or QB system. Based on convenience for calculations one system can be converted to another by either drawing the directions or by following the simple rules as given in the table-1.

Table:1: Conversion of WCB into RB

W.C.B Between	Rule for R.B	Quadrant
00 and 900	R.B = W.C.B	NE
900 and 1800	R.B = 1800 –W.C.B	SE
1800 and 2700	R.B = WCB -1800	SW
2700 and 3600	R.B = 3600-WCB	NW

The table-2 shows how the WCB of a line can be converted to a corresponding reduced bearing or QB system. The corresponding RB value and the quadrant in which it falls are provided in the table.

Table-2: Conversion of R.B into W.C.B

R.B	Rule for W.C.B	W.C.B between
NαE	W.C.B = R.B	00 and 900
SβE	W.C.B = 1800 –R.B	900 and 1800
SθW	W.C.B = 1800 + R.B	1800 and 2700
NφW	W.C.B = 3600-R.B	2700 and 3600

WCB and QB Example Problems

Before tackling more complex problems, it's important to remember that bearing angles, whether in Whole Circle Bearing (WCB) or Reduced Bearing (RB) format, can be expressed in degrees, minutes, and seconds. To perform arithmetic calculations involving bearings, it is essential to convert all angles to a single unit, typically degrees. This ensures consistent calculations unless you have access to a specialized arithmetic calculator capable of handling angles in their original format. 

Keep in mind that, 

1 degree= 60 minute = 3600 seconds

• To convert minutes to degrees, divide the minutes by 60;
• To convert the seconds to degrees, divide the seconds by 3600;

1. Convert the following Whole Circle Bearing to Quadrantal Bearing.

• 22⁰30’ 
• 211⁰54’’

Solution 1: WCB = 22⁰ 30^’

You can check the conversion table, which tells if the WCB value is between 0 and 90 degrees then the Reduced bearing value is the same. I will also explain this by depicting the angles.

If we depict this in quadrants, it lies in the first quadrant. Now, if we try to find the reduced bearing of this line OA, we can see that the reference meridian will be N and measured in the eastwards direction. Hence the bearing value of the line OA in the Quadrantal system will be the same but designated as N 22⁰30^’E.

Solution 2: WCB = 211⁰54’

You can check the conversion table. The WCB values lie between 180 and 270 degrees. The conversion is R.B = R.B = WCB -180⁰  in SW direction;

Hence, R.B = 211⁰54’ – 180⁰

54’ = 54/60 degrees = 0.9 degrees

Then R.B = (211 + 0.9) -180 = 211.9 -180 = S 31.9⁰ W

We can convert 0.9 degrees to minutes as 0.9 x 60 = 54 minutes

Therefore R.B = S 31⁰54’’ W

2. Convert the following Quadrantal Bearing S 31⁰54’’W to Whole Circle Bearing.

Solution: QB = S 31⁰54’’W

From the table, the value lies in the SW quadrant, hence the formula is WCB = 180 + RB = 180 + 31⁰54’’ = 211⁰54’’. 

If you draw the given QB value, it lies on the third quadrant as shown. The WCB value of the line must be determined that is measured from the north. From the figure, the unknown angle will be 180 + reduced bearing value, in this case, we get 211 degrees and 54 minutes as shown.

Hence, the Whole circle bearing value of S 31⁰54’’W is 211⁰54^’’.

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