Bearings
Introduction

Navigating through the streets of a city can be a simple affair. 'Take the third street on your right. Go straight over at the first set of traffic lights, and take a left at the second set onto the bridge'.

When there are no set paths to follow, navigation becomes a little more difficult. Think about finding your way through thick jungle or sailing across the ocean to a coast line that you can't even see. In such cases we need maps and some idea of distance and direction. We will be looking at just such information in this unit.
Compass points
The four main, or cardinal, compass points are north, south, east, and west. The directions between these are called north-east, south-east, south-west, and north-west. These are the directions that are represented on compasses and maps.

 Figure 1. An old map.
 Figure 2. The main compass points.

However, even eight compass points do not provide the level of precision required by many navigators. We can double the number of directions by creating more 'in-between' points. Halfway between north and north-east we have north-north-east, for example.

When we look at a compass with all these directions marked, it may seem that we have enough points to direct, say, a ship accurately. But there are many occasions when we need to refer much more accurately to the direction between two points.

To give greater directional precision, we split the compass into 360 degrees. This provides just the same accuracy that we have with
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
measurements.

 Figure 3. Ships sailing in different directions.
Two ships depart from the same port. Ship A travels in a westerly direction, ship B in a south-westerly direction, as shown in Fig.3 above. What is the difference, in degrees, between their two courses? (Hint: this is the angle x in the diagram.)
Direction using bearings
One way of using degrees with compass directions is to measure clockwise or anticlockwise from north or south. For instance, look at the following map.
 Figure 4. A part of a map.

The information that we see above can be represented in words in the following way:

'The road extends in a direction N34°W from the hotel.'

'N34°W' is shorthand for '34 degrees measured from north in a westerly direction'. When giving bearings in this way, the angle is always measured from north or south, never from east or west.

Look at the three directions and choose the correct bearings to describe them.

This is a useful way of giving directions, but it is a little more complicated than it needs to be. Why not standardize the angle which is measured and where it is measured from? In fact, this is what is usually done. Bearings are generally given as a three-figure angle that is measured clockwise from north.

 Figure 5. Examples of three-figure bearings measured clockwise from north.

So east has a
bearing
A bearing is the direction (as an angle measured clockwise from north) that a point lies from a given location. It is usually given as a three-figure bearing, e.g. 087°.
bearing
of 090°, south has a bearing of 180°, and west has a bearing of 270°.
 Figure 2. A compass.
What is the three-figure bearing of south-west?
• °
What is the three-figure bearing equivalent of south-east?
• °

Look at the radar screen below. Click on the button and use the protractor to answer the question.

You are at the point represented by the centre of the radar screen. At what bearing from you is the unidentified object?
• °

Direction and distance
Giving a direction is not always enough to direct someone to a position. Sometimes a distance is needed too. For example, if a ship needs to be directed to the site of a shipwreck, there might be no visible landmarks, so the captain would need to know how far to go in a particular direction. Fig.8 shows a survey ship that must clear the straits and arrive safely at a place designated for a geological survey. Use the directions given here to guide the ship to its destination. If you follow them accurately enough, you will be told that you have arrived at the correct position. Click on the dot representing your ship to begin the journey.

 85 km at 333°
 90 km at 029°
 95 km at 068°
 60 km at 141°
 95 km at 111°
 70 km at 134°
 45 km at 173°

 Figure 8. Using directions to guide a ship to the site of a geological survey.

If somebody travels 20 km at a bearing of 240° from a pointP to a pointQ, can we work out how they should return to their original position? If we put this information into diagrammatic form, we can find the answer.

 Figure 9. Bearing from P to Q.

Since the two north-pointing lines are
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
parallel
we know that:

Since is the bearing from Q to P, we know that the person will need to travel at a bearing of 060° for 20 km to return to P. If 240° is the forward bearing from P to Q, then 060° is known as the back-bearing from Q to P.

In the figure below the forward bearings (A to B, C to D, E to F ) are marked.

 Figure 10. Finding back-bearings.
Calculate the back-bearings in Fig.10 above.

•  The bearing from B to A is ° The bearing from D to C is ° The bearing from F to E is °

Do you notice anything about the forward- and
back-bearing
A back-bearing provides a return direction for a given bearing, e.g. 080° is the back-bearing of 260°.
back-bearing
pairs? In fact, there is always a difference of 180° between them. Construct a few examples for yourself to verify this.
Triangulation
We don't always need to start off knowing the exact distance of a position from a fixed point to be able to locate it. For instance, imagine you are trying to make a map of a dense area of rainforest.

There is rumour of a lost temple in the heart of the forest. You and your companions can see the temple's towers rising above the forest from each of the two mountain peaks that overlook the area. However, on descending into the forest to find it, the thick vegetation and uneven land make judging distance impossible and the temple remains undiscovered.

With a little thought, however, its location can be pin-pointed. Taking bearings of the position of the temple from each of the peaks and marking them on the map, we see that they cross only at one point. So this must be the site of the temple and we can mark it on our map.

 Figure 11. Bearings of the sighted temple from two points.

This process of finding an unknown position from the bearings of this point taken from two known positions is called triangulation.

Try your hand at triangulation with the following example. You are sailing a ship in dangerous waters around a rocky cape and need to find your exact position to avoid disaster. To check the readings from your instruments, you ask the two lighthouses marked on the map in Fig.12 below to give you simultaneous measurements of your bearing from their positions. These are 124° from lighthouse A and 194° from lighthouse B. Move and rotate the lines given to find out exactly which of the three marked positions is your own.

 Figure 12. Rounding the cape.
Which of the positions represents your ship?

In practice, it is a good idea to take bearings from three or more known locations, as this improves the accuracy of the triangulation.

Summary

Bearings can be given in the form N30°W (meaning 30° from north measured in a westerly direction).

Bearings are more usually given in the form 330° (meaning 330° from north measured in a clockwise direction).

For every bearing there is a back-bearing that represents the bearing required to return to the original position.

Pin-pointing an unknown position by measuring its bearing from two or more known positions is called triangulation.
Exercises
 Figure 13. Estimating bearings.
1. Look at the bearings in Fig.13 above and choose the correct answers to describe them.

• Bearing A:
Bearing B:
Bearing C:
 Figure 14. Measuring bearings.
2. Measure the angles of the bearings in Fig.14 above using the moveable protractor and write down their correct values.

•  Bearing G:  ° Bearing H:  ° Bearing I:  °

 Figure 15. Estimating bearings.
3. Look at the bearings in Fig.15 above and choose the correct three-figure answers to describe them.

• Bearing J:
Bearing K:
Bearing L:
 Figure 16. Finding bearings.
4. Find the three-figure bearings of A from B using Fig.16 above.

• Bearing 1:
Bearing 2:
Bearing 3:
5. What is the general relationship between the bearing from A to B and the bearing from B to A?
 Figure 17. Measure a bearing.
6. What is the bearing of P from Q? Use the moveable protractor.

 Figure 18. Calculating back-bearings.
7. Work out the following back-bearings.

• The bearing of A from B is ° and the bearing of C from D is °
 Figure 19. Calculating back-bearings.
8. Work out the following back-bearings.

• The bearing of E from F is ° and the bearing of G from H is °
 Figure 20. Rounding the cape.
9. Find your exact position given that you are 152° from lighthouse A and 276° from lighthouse C in Fig.20.
Well done!
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