Arab academy for science,

technology and maritime

transport

Introduction To
Navigation
Course Code (BS 131)
 Table of content

1. Navigation and Shape of the Earth 1

1.1 Navigation 1
1.2 Types of Navigation 1
1.2.1 Coastal navigation 1
1.2.2 Offshore navigation 1
1.2.3 Importance of studying coastal navigation 2
1.3 The true shape of the earth 2
1.4 Definitions 3
1.4.1 Earth's Axis 3
1.4.2 North and south poles 3
1.4.3 Great Circle 4
1.4.4 Small Circle 4
1.4.5 East and West 4
1.4.6 The Equator 4
1.4.7 Merdians 5
1.5 Earth coordinates 5
1.5.1 Latitude 6
1.5.2 Longitude 6
1.5.3 Parallel of latitude 7
1.5.4 Difference of latitude (d.lat.) 7
1.5.5 Difference of longitude (d.long.) 8
1.5.6 How to use the calculator to find d.lat. and d.long 11
1.5.7 How to find the coordinates of the arrival position 13
1.6 Units of measurement 14
1.6.1 The sea mile 14
1.6.1.1 The length of the sea mile 14
1.6.1.2 Cable 14
1.6.2 The international nautical mile 14
1.6.3 The knot 14
1.6.4 The geographical mile: 15
1.6.5 The statute mile 15
1.7 Linear measurement of latitude 15
1.8 The linear measurement of longitude 16
1.9 Rhumb line 16
1.10 Departuer 17

2. Directions on the Earth and Compasses 20

2.1 Direction 20
2.1.1 Course 20
 2.1.2 Course line 20
2.1.3 Heading 20
2.1.4 Bearing 21
2.1.5 Relative bearing 21
2.2 Compasses 24
2.2.1 Magnetic Compass 24
2.2.1.1 Magnetic Principles 24
2.2.1.2 Terrestrial Magnetism 24
2.2.1.3 Magnetic Meridian 25
2.2.1.4 Variation 25
2.2.1.5 Secular Change 26
2.2.1.6 Change in variation 26
2.2.1.7 Deviation 28
2.2.1.8 Deviation Table 30
2.2.1.9 Compass Error 30
2.2.2 Gyro Compass 32
2.2.2.1 Advantages 33
2.2.2.2 Limitations 33
2.2.2.3 Determining the gyro error 34

3. Chart Projection 39
3.1 Introduction 39
3.2 The desirable properties of preferable projection 39
3.3 The requirements of a chart appropriate for marine navigation 39
3.4 Types of Projections 39
3.4.1 Conventional Projections 39
3.4.1.1 Mollwiede’s Projection 40
3.4.2 Projections derived from the cone 40
3.4.2.1 Conical Projections 40
3.4.2.1.1 Simple conic Projection 40
3.4.2.1.2 Lambert Conformal Projection 42
3.4.2.1.3 Polyconic Projection 42
3.4.2.2 Azimuthal or Zenithal Projections 43
3.4.2.2.1 Gnomonic Projection 43
3.4.2.2.2 Stereographic Projection 44
3.4.2.2.3 Orthographic Projection 45
3.4.2.3 Cylindrical Projections 46
3.4.2.3.1 Mercator projection 46
3.4.2.3.2 Transverse Mercator projection 46
3.4.2.3.3 Oblique Mercator projection 47
3.5 Mercator Charts 48
3.5.1 The distinctive features of Mercator charts 49
 3.5.2 The limitations of Mercator charts 49
3.5.3 Meridional parts (M.P): 49
3.6 Comparison between Gnomonic Projection and Mercator projection 51

4. Position Lines 53
4.1 Introduction 53
4.2 Straight position line 53
4.2.1 True bearing 53
4.2.2 Gyro bearing 54
4.2.2.1 Gyro error 54
4.2.3 Compass bearing 56
4.2.3.1 Variation 56
4.2.3.2 Deviation 59
4.2.3.3 Total compass error 60
4.2.4 Transit bearing 63
4.2.4.1 Deviation from transit bearing 65
4.2.5 Relative bearing 66
4.2.6 Beam bearing 68
4.3 Circular position lines (Distance) 68
4.3.1 Direct distance 68
4.3.2 Indirect distance 68
4.3.2.1 Distance by Vertical Sextant Angle 69
4.3.2.2 Lighthouse first appear or disappear 70
 Chapter One
Navigation and Shape of the Earth

1.1 Navigation:

It is the art of finding the position of a ship at sea and conducting her safely from place to
place.

Fig. (1-1) The art of navigation

• Navigation is a science in that it involves the development and use of:
1. Instruments.
2. Methods.
3. Tables.
4. Almanacs.
• Navigation is an art in that it involves:
1. The proficient use of these tools.
2. The application and interpretation of information gained from such use.

1.2 Types of navigation

1.2.1 Coastal navigation:

This is the art of conducting a ship in the neighborhood of dangers, such as rocks and
shoals, and in narrow water. Fixing the ship depends on terrestrial objects such as
lighthouses, towers and masts.

1.2.2 Offshore navigation:

This is the part of navigation which is concerned with the open sea, where terrestrial
objects cannot be seen. Fixing the ship depends on electronic aids to navigation such as
global positioning system (G.P.S.), satellites as well as the observation of heavenly
bodies such as the sun, moon, planets and stars.

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1.2.3 Importance of studying coastal navigation:

To describe pilotage that is the art of handling the ship in the presence of dangers such as:
 Rocks and shoals.
 Shallow water.
 Fishing boats.
 Wrecks.
 Narrow waters.
Therefore, it is highly recommended to consider the following safety precautions while
sailing along the coast:
1. High accuracy in fixing the ship.
2. Fixing the ship at short intervals of time.
3. Safe speed.
4. Ensure that all the required charts are available and up to date.
5. The presence of up to date books related to artificial aids to navigation
such as buoys, lights, fog signals and radio aids.
6. The knowledge of charts, sailing directions and other navigational
publications.

1.3 The true shape of the earth:

The earth is formed by rotating a circle about its diameter. Although the earth is not a
perfect sphere, it may be considered so for the purposes of navigation, as the results
resulting from this assumption are always negligible.

Fig. (1-2) shape of the earth.

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 Fig. (1-3) shape of the earth.

a represents major semi axis (equatorial radius) = 3444 miles

b represents minor semi axis (polar region) = 3432 miles
Mathematical relationship to find compression C:
C = (a-b) ÷ a = (3444-3432) ÷ 3444 = 1 / 287 = 0.003

As previously mentioned, the value of compression can be neglected, however the

departures from the spherical shape are only considered when:
 Constructing charts.
 Constructing navigational books and publications.
 Sailing for a long period north or south of the equator.

1.4 Definitions:
1.4.1 Earth's Axis

Is the shortest diameter about which it rotates in space.

1.4.2 North and South Poles

The Poles are the extremities of the axis of the Earth. The earth when viewed from above
the North Pole rotates in an anti clock wise direction, the opposite pole is called the South
Pole.

3
1.4.3 The great circle:

A sphere is formed by rotating a circle about a diameter, any section of a sphere by a

plane is a circle. If the plane passes through the centre of the sphere, the resulting section
is the largest that can be obtained and is known as great circle. It is important because it
gives the sailor the shortest track between any two places which lie on it.

1.4.4 Small circle:

If the plane does not pass through the centre of the sphere, the section is known as small
circle.

Fig. (1-4) great and small circles.

1.4.5 East and west:

The direction towards which the earth rotates is called east, the opposite direction is
called west.

1.4.6 The equator:

Is the great circle midway between the poles. Its plane is perpendicular to the earth's axis
and divides the earth into North hemisphere and South hemisphere.

4
 Fig. (1-4) The equator.

1.4.7 Meridians:

These are semi great circles joining the poles and are perpendicular to the equator.

Fig.(1-5) The planes of the meridians at the polar axis, equator, north, south, east and
west.

1.5. Earth coordinates:

To find the position of any point in a plane, it is sufficient to know its shortest distances
fro two lines in that plane at right angles to each other. The corresponding axes on the
earth surface are the equator and the meridian through Greenwich. The equator is selected
because it is midway between the poles. The Greenwich meridian is selected by an
international agreement and is known as the prime meridian.

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 Greenwich
Equator

Fig.(1-6) The equator and Greenwich meridian.

1.5.1 Latitude:

The latitude of a place is the angular distance of that place north or south of the equator.
It is expressed in degrees, minutes and seconds from 0° to 90°.

Fig.(1-7) The latitude of a place.

1.5.2 Longitude:

The longitude of a place is the angular distance expressed in degrees, minutes and
seconds between its meridian and the prime meridian from 0° to 180° east or west of
Greenwich.

6
 Fig.(1-8) The longitude of a place.

1.5.3 Parallel of latitude:

This is a small circle, the plane of which is parallel to the plane of the equator.

Fig.(1-9) A parallel of latitude is parallel to the equator.

1.5.4 Difference of latitude (d.lat.):

The difference of latitude between two places F and T is the difference between the
latitudes of F and T; that is, the length FM along the meridian through F, cut off by the
parallels of latitude through F and T. This expression is used whether both latitudes have
the same sign or not.

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  If the two places are on the same side of the equator i.e. have the same sign, d.lat.
is found by subtracting the smaller latitude from the greater one.
 If the two places lie on opposite sides of the equator i.e. have different signs, d.lat.
is the sum of both latitudes.

The rule for finding d.lat. can thus be summarized as follows:

 Same names subtract
 Opposite names add.

Fig.(1-10) The difference of latitude between two places on Earth.

In northern hemisphere:
 D.lat. will be named SOUTH (S) if the ship is moving from a greater latitude
(N) to a smaller latitude (N).
 D.lat. will be named NORTH (N) if the ship is moving from a smaller latitude
(N) to a greater latitude (N).
In southern hemisphere:
 D.lat. will be named NORTH (N) if the ship is moving from a greater latitude
(S) to a smaller latitude (S).
 D.lat. will be named SOUTH (S) if the ship is moving from a smaller latitude
(S) to a greater latitude (S).
 If both latitudes have opposite names, d.lat. will have the same sign as the
arrival latitude.

1.5.5 Difference of longitude (d.long.):

The difference of longitude between two places is the length of the arc on the equator or
the angle at the pole between the meridians of the two places.
 If the two places are on the same side of the meridian of Greenwich i.e. have the
same sign, d.long.. is found by subtracting the smaller longitude from the greater
one.
 If the two places lie on opposite sides of the meridian of Greenwich i.e. have
different signs, d.lat. is the sum of both longitudes.

8
 Fig.(1-11) The difference of longitude between two places on Earth.

In Eastern hemisphere:
 D.long. will be named EAST (E) if the ship is moving from a smaller meridian
(E) to a greater meridian (E).
 D.long. will be named WEST (W) if the ship is moving from a greater meridian
(E) to a smaller meridian (E).

In Western hemisphere:
 D.long. will be named EAST (E) if the ship is moving from a greater meridian
(W) to a smaller meridian (W).
 D.long. will be named WEST (W) if the ship is moving from a smaller meridian
(W) to a greater meridian (W).

If both meridians lie on opposite sides of the prime meridian:

1. If the ship is steaming from a longitude EAST (E) to a longitude WEST(W),
d.long. will be named WEST (W).
2. If the ship is steaming from a longitude WEST (W) to a longitude EAST(E),
d.long. will be named EAST (E).

Remarks:

A slight complication may appear if the sum of the two longitudes is greater than
180°, in this case the d.long. will be 360° minus the sum of the two longitudes.
Consequently, the sign of d.long. will be changed.

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 Miscellaneous examples to find d.lat and d.long between two places:
Find the d.lat and d.long between the following places:

1- A (30° 00`N, 020° 00`E) B (40° 00`N, 050° 00`E)

Lat. A 30° 00`N Long. A 020° 00`E

Lat. B 40° 00`N Long. B 050° 00`E
d.lat. 10° 00`N d.long. 010° 00`E

2- A (35° 00`S, 040° 00`W) B (60° 00`S, 070° 00`W)

Lat. A 35° 00`S Long. A 040° 00`W

Lat. B 60° 00`S Long. B 070° 00`W
d.lat. 25° 00`S d.long. 030° 00`W

3- A (60° 20.7`N, 080° 34.3`E) B (40° 47.9`N, 050° 52.7`E)

Lat. A 60° 20.7`N Long. A 080° 34.3`E

Lat. B 40° 47.9`N Long. B 050° 52.7`E
d.lat. 19° 32.8`S d.long. 029° 41.6`W

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4- A (65° 43.5`S, 120° 13.4`W) B (39° 57.8`S, 102° 48.3`W)

Lat. A 65° 43.5`S Long. A 120° 13.4`W

Lat. B 39° 57.8`S Long. B 102° 48.3`W
d.lat. 25° 45.7`N d.long. 017° 25.1`E

5- A (43° 16.7`S, 130° 42.5`W) B (18° 46.4`N, 083° 53.4`E)

Lat. A 43° 16.7`S Long. A 130° 42.5`W

Lat. B 18° 46.4`N Long. B 083° 53.4`E
d.lat. 62° 03.1`N d.long. 214° 35.9`E
- 360°
d.long. 145° 24.1W

6- A (28° 20`N, 040° 15`E) B (37° 50`S, 018° 40`W)

Lat. A 28° 20`N Long. A 040° 15`E

Lat. B 37° 50`S Long. B 018° 40`W
d.lat. 66° 10`S d.long. 058° 55`W

1.5.6 How to use the calculator to find d.lat. and d.long :

 To find d.lat. by using a calculator, we use the following relationship:

D=B–A
Where:
B = arrival latitude
A = departure latitude
D = d.lat.

Data will be entered as follows:

 Enter northern latitudes either with a positive sign (+) or without any signs.
 Enter southern latitudes with a negative sign(-).
 If the result is positive(+), d.lat. will be named NORTH (N).
 If the result is negative(-), d.lat. will be named SOUTH (S).

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 °

°
Lat.B 00 Lat.B 00 ' _ Lat.A 00 ° Lat.A 00 ' = D.lat. 00 00'

Example 1:

Calculate the d.lat. between lat. 37° 22.7` N and lat. 25° 52.3` N.

25 ° 52.3 ' _ 37 ° 22.7 ' = - 11° 30.4 '

The negative sign means that d.lat. is SOUTH

D.lat. = 11° 30.4` S.

Example 2:

Calculate the d.lat. between lat. 12° 34.8` N and lat. 05° 18.4` S.

- 05 ° 18.4 ' _ 12 ° 34.8 ' = - 18° 02.2 '

D.lat. = 18° 02.2` S.

 To find d.long. by using a calculator, we use the following relationship:

D=B–A
Where:
B = arrival longitude.
A = departure longitude.
D = d.long.

Data will be entered as follows:

 Enter eastern meridians either with a positive sign (+) or without any signs.
 Enter western meridians with a negative sign(-).
 If the result is positive(+), d.long. will be named EAST (E).
 If the result is negative(-), d.long. will be named WEST (W).
°
Long.B 00 ' _ Long. A 00° Long.A 00 ' = D.long. 00 00 '
°

Long.B 00

Example 3:

Calculate the d.long. between long. 078° 12.4` E and long. 068° 10.4` E.

12
Solution:
°
10.4 ' _ 078 ° 12.4 ' = - 010 02.0 '

°
068

D.long. = 010° 02.0` W.

Example 4:

Calculate the d.long. between long. 174° 32` W and long. 171° 15` E

Solution:
°
15.0 ' _ - 174° 32.0 ' = 345 47 '
°

171

Since d.long. is greater than 180°, subtract the answer from 360° and reverse its sign.
D.long. = 360° - 345° 47` E
D.long. = 014° 13`

1.5.7 How to find the coordinates of arrival position ( latitude and

longitude) by knowing initial position, d.lat. and d.long.

1. if both the initial latitude and d.lat. have the same sign, ADD (+) to find the
arrival latitude B, latitude B will have the same sign as both the initial latitude
and d.lat.
2. if the initial latitude and d.lat. have the opposite signs, SUBTRACT (-) to find the
arrival latitude B, latitude B will have the sign of either the initial latitude or d.lat.
whichever is greater.
3. if both the initial longitude and d.long. have the same sign, ADD (+) to find the
arrival longitude B, longitude B will have the same sign as both the initial
longitude and d.long.
4. if the initial longitude and d.long. have the opposite signs, SUBTRACT (-) to find
the arrival longitude B, longitude B will have the sign of either the initial
longitude or d.long. whichever is greater.

B=D+A
1.6. Units of measurement

1.6.1 The sea mile:

This is the length of 1` of arc measured along the meridian, in the latitude of the position,
its length varies both with the latitude and with the figure of the earth in use. The sea mile
is used for the scale of the latitude on large scale admiralty charts as distances are
measured using the latitude graduations of the chart borders and is denoted (M).

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 Fig.(1-12) The sea mile.

In fig.(1-12), M is a place on the surface of the earth, C being the centre of

curvature of the place at M and AMB is the arc measured along the meridian which is
equivalent to 1` at C. AMB represents the sea mile at M.

1.6.1.1 The length of the sea mile:

The length of the sea mile varies from 1842.9 meters at the equator to 1861.7 meters at
the pole. The mean value of the sea mile at latitude 45° is 1852.3 meters.

1.6.1.2 Cable:

The cable is one tenth of the sea mile (0.1of sea mile), the length of the cable varies from
184.3 meters to 186.2 meters. The cable is of a great convenience to coastal navigation
specially while approaching ports, maneuvering and mooring.

1.6.2 The international nautical mile:

This is a standard fixed length of 1852 meters and is denoted by (n.miles).

1.6.3 The knot:

In navigation, the unit of speed is one nautical mile per hour and that unit is called the
knot.

1.6.4 The geographical mile:

This is the length of one minute of arc, measured along the equator and is equal to 1855.4
meters.

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 Fig.(1-13)The geographical mile.

1.6.5 The statute mile:

This is a unit of measurement of 1760 yards or 5280 feet.

1.7 Linear measurement of latitude:

Linear measurement of latitude is the length of the arc on the meridian between the
equator and the latitude of the place and is measured in sea miles north or south of the
equator.

Fig.(1-14) Linear measurement of latitude.

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 In fig.(1-14), the latitude of point M is 60° N, so angle MLW will be equal 60°

The length of the arc MW = 60° × 60` = 3600`

Similarly, if the point Mı is at a distance of 1800 miles south of the equator

Latitude of Mı = 1800 ÷ 60 = 30° S.

1.8 The linear measurement of longitude:

The distance on the earth’s surface between any two meridians is greatest at the equator
and diminishes uniformly until it is zero at the poles where all the meridians meet. The
linear distance of a degree of longitude on the surface of the earth therefore varies in
accordance with its latitude and cannot be taken as a standard measure of length. For
instance, the distance on the earth’s surface representing 30° of longitude at latitude 60°
N. is 902½ nautical miles, whereas at the equator it is 1800 nautical miles.

Fig.(1-15) Linear measurement of longitude.

1.9 Rhumb line:

A line on the earth’s surface which cuts all meridians at the same angle is called a rhumb
line. In fig.(1-16) FABCT is the rhumb line joining F to T. the angles PFA, PAB, PBC,
and PST are all equal and any one of them may be taken as the course.

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 Fig.(1-16) Rhumb line.

1.10 Departure:

Departure is the distance made good in an east-west direction in sailing from one place to
another along a rhumb line and is measured in nautical miles.

Fig.(1-17) Departure.

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 Questions

1. What are the types of navigation, mention the importance of studying coastal
navigation.

2. Describe the true shape of earth then mention the mathematical relationship that
rules the compression of earth. When is this compression considered?

3. Define the following: Great circle, Small circle, Equator, Parallel of latitude,
Meridian, Difference of latitude, Difference of longitude.

4. Calculate the d.lat. and fill the following table when sailing from A to B:
No. Lat.A Lat.B D.lat. Sign
1 25° 00` N 45° 00`N
2 53° 00` S 68° 00`S
3 12° 17` N 23° 18`N
4 48° 36` S 08° 19`S
5 01° 16.5` S 03° 08.4`N
6 19° 42` N 11° 15`S
7 04° 32` N 19° 14`N
8 50° 20` N 30° 42`N
9 60° 13` S 48° 17`S
10 40° 50` N 17° 40`S

5. Calculate the d.long. and fill the following table when sailing from A to B:

No. Long. A Long. B D.long. Sign

1 070° 18` E 037° 20`W
2 080° 44` W 37° 14`E
3 018° 52` E 072° 32`W
4 120° 18` E 027° 40`W
5 170° 13` W 150° 17`E
6 030° 14` E 085° 15`W
7 003° 28` W 002° 50`E
8 069° 48` W 007° 17`W
9 034° 19` E 020° 18`W
10 118° 45` E 115° 30`E

6. Calculate d.lat. and d.long. for the following:

 A ( 50° 30`N, 020° 15` E) and B (70° 20`N, 045° 18`E).

 A ( 56° 12`N, 112° 20` E) and B (40° 43`N, 080° 50`E).
 A ( 30° 30`S, 060° 50` W) and B (48° 50`N, 080° 30`E).
 A ( 25° 25`S, 045° 30` W) and B (18° 52`S, 028° 42`E).
 A ( 63° 47`S, 120° 39` E) and B (80° 26`S, 090° 40`W).

18
7. Find the coordinates of arrival position:

 A (45° 17` N, 113° 27` W), d.lat. = 25° 13` N, d.long. = 014° 33`E.
 A (35° 25` N, 179° 24` E), d.lat. = 29° 05` N, d.long. = 024° 18`E.
 A (15° 23` N, 045° 35` W), d.lat. = 45° 27` S, d.long. = 073° 27`W.

8. Define the following: Sea mile, Cable, International nautical mile, Knot,
Geographical mile, Statute mile.

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 Chapter Two
Directions on the Earth and Compasses

2.1 Direction

Is the position of one point relative to another. Navigators express direction as the angular
difference in degrees from a reference direction, usually north or the ship’s head.

2.1.1 Course

Is the horizontal direction in which a vessel is intended to be steered, expressed as

angular distance from a north clockwise through 360. The term applies to direction
through the water, not the direction intended to be made good over the ground.
The course is often designated as:
 True Course: is the angular distance from true north clockwise through 360
 Magnetic Course: is the angular distance from magnetic north clockwise through
360
 Compass Course: is the angular distance from compass north clockwise through 360
 Gyro course: is the angular distance from the gyro compass north clockwise through
360

2.1.2 Course line

Is a line drawn on a chart extending in the direction of a course. It is sometimes

convenient to express a course as an angle from either north or south, through 90or
180. In this case, it is designated course angle (C) and should be properly labeled to
indicate the origin and direction of measurement. Thus:

C N35E = Cn 035 (000+ 35)

C N155W = Cn 205 (360- 155)
C S47E = Cn 133 (180- 47)

2.1.3 Heading

Is the direction in which a vessel is pointing at any given moment, expressed as angular
distance from 000clockwise through 360. It is easy to confuse heading and course.
Heading constantly changes as a vessel yaws back and forth across the course due to sea,
wind, and steering error.

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 Figure (3-1) Course line, track, track made good, and heading

2.1.4 Bearing

Is the direction of one terrestrial point from another, expressed as an angle from 000
(North) clockwise to 360. A bearing or (position line) as it will be mention in chapter 4,
can be designated as:
True bearing: (if it is referred to the true north)
Gyro bearing: (if it is referred to the gyro north)
Magnetic bearing: (if it is referred to the magnetic north)
Compass bearing: (if it is referred to the compass north)
Note: dealing with these types of bearing will be discussed shortly.

2.1.5 Relative Bearing

A relative bearing is measured relative to the ship’s heading from 000 (dead ahead)
clockwise through 360.
To convert a relative bearing to a true bearing, add the true course.

True Bearing = True Course + Relative Bearing.

Example:

Given the true course 170° T and the relative bearing 250° find the true bearing

Solution:

True Bearing = True Course + Relative Bearing

True Bearing = 170° + 250° = 420° - 360° = 060° T.

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 Figure (3-2) Relative Bearing

However, relative bearing is sometimes measured right or left from 000at the ship’s
head through 180

If the relative bearing is measured to the right from the ship’s head (green) through 180
the true bearing will be calculated as follows:

True Bearing = True Course + Relative Bearing.

Example:

Given the true course 060° T and the relative bearing 125° green, find the true bearing

Solution:

True Bearing = True Course + Relative Bearing

True Bearing = 060° + 125° = 185° T.

If the relative bearing is measured to the left from the ship’s head (red) through 180 the
true bearing will be calculated as follows:

True Bearing =True Course - Relative Bearing.

22
Example:

Given the true course 315° T and the relative bearing 065° red, find the true bearing

Solution:

True Bearing =True Course - Relative Bearing.

True Bearing = 315° - 065° = 250° T.

Exercise
1. The heading of a ship is 170° T. The relative bearing of a distant tower is 155°. What is the
true bearing?

2. Given the ship’s true course is 130° T. The relative bearing is 060°. What is the
true bearing?

3. The true course of a ship is 315° T. The relative bearing of another ship is 125°.
Find the ship’s true bearing.

4. A ship’s true course is 090° T and the relative bearing of a lighthouse is 045°
green. What is the true bearing?

5. A ship’s true course is 330° T and the relative bearing of a distant object is 030°
red. What is the true bearing?

23
2.2 Compasses

There are two types of compasses in the maritime filed:

1. Magnetic compass
2. Gyro compass

2.2.1 Magnetic compass

The basic mechanism of modern magnetic compasses is the same as that of the very
earliest ones used, a small bar magnet freely suspended in the magnetic field of the earth.
Refinements have been added for greater accuracy, steadiness of indication, and ease
reading, but the basic mechanism remains unchanged.

Figure (3-3). The Magnetic Compass

2.2.1.1 Magnetic principles:

Magnetism is the property of a certain metals to attract or repel items. The space around
each magnet in which its influence can be detected is called its field which can be
pictured as being composed of many lines of forces. Each magnet has two opposite
polarity one is termed north and the other south.

2.2.1.2 Terrestrial Magnetism

Consider the Earth as a huge magnet surrounded by lines of magnetic flux connecting its
two magnetic poles. These magnetic poles are near, but not coincidental with, the Earth’s
geographic poles. The angular difference between the true meridian and the magnetic
meridian is called variation. This variation has different values at different locations on
the Earth. These values of magnetic variation may be found on pilot charts and on the
compass rose of navigational charts. The poles are not geographically static. They are
known to migrate slowly, so that variation for most areas undergoes a small annual
change, the amount of which is also noted on charts.

24
 Figure (3-5). Terrestrial magnetism

2.2.1.3 Magnetic meridian

At the surface of the earth, the lines of force become magnetic meridians. These are
irregular lines which cannot be printed on charts covering large areas; their irregularity is
primarily caused by the non-uniform distribution of magnetic material in the earth.

2.2.1.4 Variation

The difference at any location between the directions of the magnetic and true meridians
is the variation. It is called easterly (E) if the compass needle, aligned with the magnetic
meridian, points eastward or to the right of true north, and westerly (W), if it points to the
left. The magnetic variation and its annual change are shown on charts, so that directions
indicated by the magnetic compass can be corrected to true directions. Since variation is
caused by the earth's magnetic field, its value changes with the geographic location of the
ship, but is the same for all headings of the ship.

25
 True
True
North
North

Magnetic
Magnetic
North
North

var.
var.

Easterly Variation
Westerly Variation

Figure (3-6). Easterly and Westerly Variations

2.2.1.5 Secular change

The earth's magnetic field is not constant in either intensity or direction. The changes are
diurnal (daily), yearly, and secular (occurring over a longer period of time). The change
generally consists of a reasonably steady increase or decrease in the variation. This
change may continue for many years, sometimes reaching large values, remain nearly
stationary for a few years, and then reverse its trend.

2.2.1.6 Change in variation

If the change of inclination of the magnetic meridian to the true meridian is measured
over a period of several years at a given location, its future value for the next few years
can be predicted with considerable accuracy. Charts generally indicate the values of the
variation for a stated year, and note the annual amount and direction of the secular
change, so that the value for any subsequent year, within a reasonable period, may be
calculated. This annual change is printed within the compass rose on the chart.

26
 Figure (3-7). The Compass Rose

How to convert from magnetic course to true course and vice versa:
In order to convert from magnetic course to true course, add easterly errors, or simply,
correcting add east. When applying this rule, it is necessary to consider a magnetic
direction as the "least correct" expression of direction as it contains one error, variation.
Example:
A ship’s magnetic course is 155°. For this heading the variation is 5° E. Find the ship’s
true course.
Solution:
Since the variation is easterly, it must be added to find the true course. Hence the true
course is:

27
True Course = Magnetic Course + Variation
True Course = 155° + 5° = 160° T
Example:
A ship’s magnetic course is 330°. For this heading the variation is 12° W. Find the ship’s
true course.
Solution:
Since the variation is westerly, it must be subtracted to find the true course. Hence the
true course is:
True Course = Magnetic Course - Variation
True Course = 330° - 12° = 318° T
2.2.1.7 Deviation

A compass needle free to turn horizontally tends to align itself with the earth's magnetic
lines of force. Unfortunately, it is not free to do so in a steel ship; such ships have marked
magnetic properties of their own, and these tend to deflect the compass from the magnetic
meridian. The divergence thus caused between the north-south axis of the compass card
and the magnetic meridian is called deviation (Dev. or D).

The possibility of deviation from electrical circuits must be considered. Direct currents
flowing in straight wires establish magnetic fields. Care must be taken that all wiring near
a compass is properly installed to eliminate or reduce any effect on the compass; checks
must be made for deviation with the circuits turned on and off.
If no deviation is present, the compass card lies with its axis in the magnetic meridian and
its north point indicates the direction of magnetic north.
1. If deviation is present and the north point of the compass points eastward of
magnetic north, the deviation is named easterly and marked E.
2. If it points westward of magnetic north, the deviation is named westerly and
marked W.

The navigator can easily find the correct variation by referring to the chart of his locality.
Deviation varies with changes in the ship's heading. In addition, it often changes with
large changes in the ship's latitude.

28
 Figure (3-8). Westerly variation and deviation

Figure (3-9). Westerly variation and easterly deviation

29
2.2.1.8 Deviation tables:

The deviation on various headings is tabulated on a form called deviation table and
posted near the magnetic compass. If it is desired to find the deviation when a compass
heading is given, enter the table with the ship’s compass course and extract the deviation
for this specific compass course. It should be noted that the deviation tables tabulate
deviation for compass courses.

Compass course Deviation Compass course Deviation

000° 4.0° E 180° 3.2° E
010° 3.0° E 190° 2.2° E
020° 2.0° E 200° 1.2° E
030° 0.4° E 210° 0.5° E
040° 1.2° W 220° 1.3° W
050° 2.3° W 230° 2.3° W
060° 3.2° W 240° 3.3° W
070° 3.2° W 250° 4.3° W
080° 5.4° W 260° 3.4° W
090° 4.3° W 270° 2.5° W
100° 3.2° W 280° 1.4° W
110° 2.3° W 290° 0.4° W
120° 1.3° W 300° 0.4° E
130° 0.6° W 310° 1.0° E
140° 1.4° E 320° 1.3° E
150° 2.3° E 330° 2.0° E
160° 3.3° E 340° 3.0° E
170° 4.3° E 350° 4.0° E

Deviation table.
2.2.1.9 Compass error

The algebraic sum of variation and deviation is compass error. The navigator must
understand thoroughly how to apply variation, deviation, and compass error, as he is
frequently required to use them in converting one kind of direction to another.
From the foregoing, it should be apparent that there are three ways in which a direction
can be expressed. We can express any given direction, if we understand that:

 True differs from magnetic by variation.

 Magnetic differs from compass by deviation.
 Compass differs from true by compass error.

30
A summary of heading relationships follows:

1. Deviation is the difference between the compass heading and the magnetic
heading.
2. Variation is the difference between the magnetic heading and the true heading.
3. The algebraic sum of deviation and variation is the compass error.

Rules for applying compass errors

The following simple rules will assist in correcting and uncorrecting the compass:

Compass least, error east

Compass best, error west.

1. When correcting (converting from compass to true):

 Add easterly errors
 Subtract westerly errors (Remember: “Correcting Add East”).
2. When uncorrecting (converting from true to compass):
 Subtract easterly errors
 Add westerly errors

Example 1:

A ship is heading 127° per standard compass. For this heading, the deviation is 16° E and
the variation is 4° W in the area.
Required:
1. The magnetic heading
2. The true heading

Solution:

Since the deviation is easterly, it must be added.

The magnetic heading is 127° + 16° = 143°.
Since the variation is westerly, it is subtracted.
The true heading is 143° - 4° = 139°.
In this case the compass error is 16° E - 4° W = 12° E. Applying this directly to the
compass heading, we find the true heading is 127° + 12° = 139°, as previously
determined.

Example 2:

A ship's course is 347° C. The deviation is 4° W and the variation is 12° E.

31
Required:
1. The magnetic course
2. The true course

Solution

The deviation is subtracted and the magnetic course is 347° - 4° = 343°.

The variation is added and the true course is 343° + 12° = 355°.

Example 3:

A ship's course is 009° C. The deviation is 2° W and the variation is 19° W.

Required:
1. The magnetic course
2. The true course

Solution:

The magnetic course is 009° - 2° = 007°.

The true course is 007° - 19° = 348°. Since 000° is also 360°, this is the same as 367°-
19° = 348°.

Example 4:

From a chart, the true course between two places is found to be 221°. The variation is 9°
E and the deviation is 2° W.
Required:
1. The magnetic course
2. The compass course

Solution:

It is necessary to incorrect; the easterly variation is subtracted and the westerly deviation
is added.
The magnetic course is 221° - 9° = 212° C
The compass course is 212° + 2° = 214° M

2.2.2 Gyro compass

In the search for an instrument which would indicate true north rather than magnetic
north, the gyrocompass was developed early last century. The gyrocompass inherently is
capable of oscillating about its vertical or azimuth-indicating axis. The gyro is used
increasingly in navigation aboard ships today. A gyrocompass consists of one or more
north-seeking gyroscopes with suitable housing, power supply, etc. It must have a scale
for reading direction and usually has some method for electrically detecting direction and
transmitting this information as signals to other equipment.

32
 Figure (3-10). The Gyro Compass

2.2.2.1 Advantages

The gyrocompass has the following advantages over the magnetic compass:
1. It seeks the true meridian instead of the magnetic meridian.
2. It can be used near the earth's magnetic poles, where the magnetic compass is
useless.
3. It is not affected by surrounding magnetic material which might seriously reduce
the directive force of the magnetic compass.
4. If an error exists, it is the same on all headings, and correction is a simple process.
5. Its information can be fed electronically into automatic steering equipment, course
(DR) recorders, and inertial navigation systems.

2.2.2.2 Limitations

In spite of the many advantages and undoubted capabilities of a modern gyrocompass,

there are certain disadvantages inherent in its design:
1. It requires a constant source of electrical power.
2. It requires intelligent care and attention if it is to give the kind of service of which
it is inherently capable.
3. The accuracy decreases when latitudes above 75 degrees are reached.
4. If operation is interrupted for any length of time long enough for it to become
disoriented, a considerable period of time, as much as four hours, may be required
for it to settle back into reliable operation.

33
2.2.2.3 Determining the gyro error

By any one of several methods, it is a relatively easy process for a navigator to determine
the numerical value of the gyro error using simple arithmetic. The difficulty arises in
determining the label of the error low (east) or high (west). A simple memory-aid phrase
can be used as before:
1. If the gyro reading is greater than the true reading, the error will
be labeled high.
2. If the gyro reading is smaller than the true reading, the error will
be labeled low.

Rules for applying gyro errors

1. When correcting (converting from gyro to true):
 Add low errors
 Subtract high errors
2. When uncorrecting (converting from true to gyro):
 Subtract low errors
 Add high errors

True True
North North
Gyro
Gyro North
North

Gyro
Error
Gyro
Error

Gyro Error HIGH Gyro Error LOW

34
Example 1:

Two beacons in line are sighted with a gyrocompass repeater, and found to be bearing
136.5° per gyrocompass. According to the chart, the bearing of these beacons when in
line is 138° true.
Required:
The gyro error (G.E)

Solution:

Numerically, the gyro error is the difference between gyro and true bearings of the
objects in range
The gyro error is 138°- 136.5°= 1.5°.
Since the gyro bearing is smaller than the true bearing, the error is low.
Answer:
G.E. = 1.5° L.

Example 2:

A light ashore is sighted, and by gyrocompass repeater is observed to bear 310.0° per
gyrocompass. From the ship's fixed position, the charted true bearing of the light is
measured as 308.5° true.
Required:
The gyro error (G.E)

Solution:

As before, the gyro error is the difference between the gyro and the true bearing, or:
The gyro error is310°- 308.5°= 1.5°
Since the gyro bearing is greater than the true bearing, the error is high.

Answer:
G.E. = 1.5° H.

Example 3:

A ship is heading 130° per gyrocompass (G.H). The gyro error (G.E) is 1° L.
Required:
The true heading (T.H)

Solution:

Since error is low, it must be added.

The true heading is 130° + 1° = 131°.
Answer:
T.H = 131° T

35
Example 4:

A ship is heading 020° per gyrocompass. The gyro error is 1° H.

Required:
The true heading

Solution:

Since the error is high, it must be subtracted.

The true heading is 020° - 1° = 019°
Answer:
T.H = 019° T

Example 6:

From a chart, the true course between two places is found to be 151°; the G.E is 1° L.
Required:
The heading per gyrocompass to steer 151° true

Solution:

Since low errors are added to gyro to obtain true, they must be subtracted when
converting from true to gyro, or 151° - 1° = 150°.
Answer:
G.H = 150° g

36
 Exercise
1. Convert from Gyro course / bearing to True course / bearing:

Gyro course/ bearing Gyro error True course/ bearing

018° g 2° H
120° g 3° L
345° g 6° H
240° g 8° L
358° g 3° L

2. Convert from True course / bearing to Gyro course / bearing:

True course/ bearing Gyro error Gyro course/ bearing

058° T 2° H
142° T 3° L
330° T 4° H
178° T 5° L
358° T 6° L

3. Calculate the gyro error:

True course/ bearing Gyro course/ bearing Gyro error

058° T 062° g
142° T 139° g
330° T 334° g
178° T 175° g
358° T 360° g

4. Calculate the total error for the following:

Variation Deviation Compass error

5° E 2° E
14° W 13° E
3° E 9° W
8° W 5° W
13° E 3° E

37
 5. Calculate the True courses for the following Compass courses:

Compass Course Variation Deviation Compass True Course

error
235° C 5° E 2° E
064° C 14° W 13° E
317° C 3° E 9° W
135° C 8° W 5° W
035° C 13° E 3° E

6. Calculate the Compass courses for the following True courses:

True Course Variation Deviation Compass error Compass

Course
219° T 15° W 10° E
347° T 1° W 3° W
095° T 3° E 2° E
118° T 8° E 5° W
055° T 4° W 3° E

7. Define variation and deviation.

8. What does the true north and the magnetic north mean?

9. What are the advantages and limitations of the gyrocompass?

10. A light bore 210.0° by the gyrocompass. The true bearing of the light was 208.0°
T. What is the gyro error?

11. The true course between two places is found to be 310° T; the G.E is 1° L. What
is the gyro course?

12. The gyro course between two places is found to be 055°; the G.E is 2° H. What is
the true course?

13. A distant terrestrial object bore 135° by the gyrocompass. The G.E is 4° L. What
is the true bearing of this object?

38
 Chapter Three
Chart Projections

3.1 Introduction:

In the practice of navigation, the navigator requires drawings of the earth surface on
which to lay off his proposed course, fix the position of his ship and find where he is in
relation to the land. The problem is to show part of the surface of the sphere which has
three dimensions as a flat surface which has only two dimensions. The sphere cannot be
unrolled into a plane surface as can a cylinder or a cone. Distortion is therefore inevitable
when a flat drawing of its surface is made.

3.2 The desirable properties of preferable projection:

 True shape of physical features

 Correct angular representation.
 Equal area or the representation of areas in their correct relative proportions
 Constant scale values for measuring distances
 Great circles represented as straight lines
 Rhumb lines represented as straight lines

3.3 The requirements of a chart appropriate for marine navigation:

 Plane surface.
 Keeps the true shape of physical features.
 Rhumb lines appear as straight lines.
 Measure courses and bearings.
 Measuring distances on the rhumb line.
 Fixing the ship.

3.4 Types of Projections:

 Conventional projections.
 Projections derived from the cone.

3.4.1 Conventional Projections:

This type of projections is based on mathematical and geometrical rules to fulfill certain
requirements, the most famous type of conventional projections is the Mollwiede
projection.

39
3.4.1.1 Mollwiede’s Projection:

This is an equal area projection where the parallels of latitude appear as straight lines. It
is frequently used for world maps showing distributions. All parallels of latitude are
straight lines and all meridians are semi-ellipse. The central meridian is a straight line and
those 90° east and west take the shape of semi-circles.

Fig. (2-1) Mollwiede’s Projection

3.4.2 Projections derived from the cone:

This kind of projections is based on the idea that there is a cone surrounding the earth.
This main type is divided into three types:
1. Conical projection.
2. Azimuthal projection.
3. Cylindrical projection.

3.4.2.1 Conical Projections:

1. Simple conic projection.

2. Lambert conformal projection.
3. Polyconic projection.

3.4.2.1.1 Simple conic Projection:

The cone rests on the sphere and touches the sphere along a parallel of latitude. This
parallel is known as the standard parallel. When it is spread out flat to form a map, the
pole appears as an incomplete circle. The meridians are straight lines converging at the
pole and parallels of latitude are concentric circles. The standard parallel appears as an
arc of a circle with its centre at the apex of the cone. The error increases as the distance
from the standard parallel increases.

40
 Fig. (2-2) A simple conic projection.

Fig. (2-3) A simple conic map of the Northern Hemisphere.

41
3.4.2.1.2 Lambert Conformal Projection:

The useful latitude range of the simple conic projection can be increased by using a
secant cone intersecting the earth at two standard parallels one towards the north and the
other towards the south of the area to be taken. The area between the two standard
parallels is compressed, and that beyond is expanded. Such a projection is called conic
projection with two standard parallels. This kind of projection is used in aero-navigation
charts.

Fig. (2-4) A conic projection with two standard parallels.

3.4.2.1.3 Polyconic Projection:

This projection is obtained by using a series of cones; each parallel is constructed as if it

were the standard parallel of a simple conic projection. Parallels appear as nonconcentric
circles; meridians appear as curved lines converging toward the pole and concave to the
central meridian.

42
 Fig.(2-5) a polyconic map of north America.

3.4.2.2 Azimuthal or Zenithal Projections:

If points on the earth are projected directly to a plane surface, a map is formed at once,
without cutting and flattening. There are three main types of azimuthal projection.
 Gnomonic Projection.
 Stereographic Projection.
 Orthographic Projection.

Fig. (2-6) Azimuthal projections: A, gnomonic; B, stereographic; C, (at infinity)

orthographic.

3.4.2.2.1 Gnomonic Projection:

If a plane is tangent to the earth, and points are projected geometrically from the center of
the earth, the result is a gnomonic projection. It can be demonstrated by placing a light at
the center of a transparent terrestrial globe and holding a flat surface tangent to the
sphere.

Fig.(2-7) Gnomonic projection.

43
The main features of Gnomonic Projection:

1. Great circles appear as straight lines on the chart.

2. Rhumb lines appear curved.
3. Meridians are straight lines converging towards the nearer pole.
4. Parallels of latitude are curves
5. The equator appears as a straight line.

The distinctive features of Gnomonic charts:

1. Great circles appear as straight lines on the chart.

The limitations of Gnomonic charts:

1. Rhumb lines appear curved.

2. Distance scale changes with the latitude.
3. Incorrect angular relationship.
4. Unequal area representation.

Fig. (2-8) oblique gnomonic projection. Fig. (2-9) oblique gnomonic map.

3.4.2.2.2 Stereographic Projection:

A stereographic projection results from projecting points on the surface of the earth onto
a tangent plane, from a point on the surface of the earth opposite the point of tangency.
Great circles through the point of tangency appear as straight lines. Other circles such as
meridians and parallels appear as either circles or arcs of circles except for those passing
through the tangent point. The stereographic projection can show an entire hemisphere
without excessive distortion.

44
 Fig. (2-10) Stereographic projection.

Fig.(2-11) A stereographic map of the Western Hemisphere.

3.4.2.2.3 Orthographic Projection:

If terrestrial points are projected geometrically from infinity to a tangent plane, an

orthographic projection results. Its principal use is in navigational. The meridians would
appear as ellipses, except that the meridian through the point of tangency would appear as
a straight line and the one 90 away would appear as a circle.

Fig.(2-12) An equatorial orthographic projection. Fig.(2-13) orthographic map of W. Hemisphere.

45
3.4.2.3 Cylindrical Projections:

Cylindrical projections are special conical projections, using height of infinity, this means
that the angle at the apex of the cone is zero and the cone becomes a cylinder that
surrounds the earth. There are three main types of cylindrical projections:
1. Mercator projection.
2. Transverse Mercator projection.
3. Oblique Mercator projection.

3.4.2.3.1 Mercator projection:

Navigators most often use the plane conformal projection known as the Mercator
projection. The Mercator projection is based upon the idea that there is a cylinder
touching the Earth along the equator. It is used in the construction of all nautical charts
whose scale is less than (1/5000).

Fig. (2-14) a cylindrical projection.

3.4.2.3.2 Transverse Mercator projection:

In this type of projection, the cylinder touches the Earth along opposing longitude lines.
Since the area of minimum distortion is near a meridian, this projection is useful for
charts covering a large band of latitude and extending a relatively short distance on each
side of the tangent meridian. The meridians and parallels appear as curved lines while the
equator and the two tangents appear as straight lines.

46
 Fig. (2-15) Transverse Mercator projection.

Fig. (2-16) a transverse Mercator map of the western hemisphere.

3.4.2.3.3 Oblique Mercator projection:

If the cylinder does not contact the Earth along north-south lines or along the Equator,
then the projection is “Oblique Cylindrical”. This projection is used principally to depict
an area in the near vicinity of an oblique great circle.

47
 Fig. (2-17) an oblique Mercator projection.

3.5. Mercator charts:

If a light is placed at the centre of a transparent terrestrial globe, a cylinder is placed

around the earth, tangent along the equator, and the planes of the meridian are extended,
they intersect the cylinder in a number of vertical lines. If the cylinder is cut along a
vertical line (a meridian) and spread out flat, a plane surface of the earth will be obtained.
Meridians appear as straight lines perpendicular to the equator and extending to infinity.
The longitude scale is constant in all latitudes. The scale of latitude and distance at any
part of a Mercator chart is proportional to the secant of the latitude for that part. For this
reason, the amount of distortion in any latitude is governed by the secant of that latitude.

48
 Since the projection is conformal, expansion is the same in all directions and angles are
correctly shown. Rhumb lines appear as straight lines, the directions of which can be
measured directly on the chart. Distances can also be measured directly if the spread of
latitude is small. Great circles, except meridians and the equator, appear as curved lines
concave to the equator. Small areas appear in their correct shape but of increased size
unless they are near the equator.
 Greenland (lat.70° N) appears as broad as Africa is drawn at the equator
although Africa is three times as broad as Greenland (sec.70° = 3).
 Borneo, an island on the equator appears about the same size as Iceland
(lat.65° N) although Borneo is about five and a half times as large as
Iceland.

To overcome the distortion in shape, both the meridians and parallels are expanded at
the same ratio with the increased latitude. The expansion is equal to the secant of the
latitude, with a small correction for the ellipticity of the earth. Since the secant of 90is
infinity, the projection cannot include the poles.

Remarks:
1. Mercator charts are graduated along the left and right hand edges for
latitude and distance.
2. The longitude scale (along the top and bottom) is used only for laying
down or taking off the longitude of a place but never for measuring
distances.

3.5.1 The distinctive features of Mercator charts:

 Parallels of latitude and meridians appear as straight lines.

 Rhumb line appears as a straight line.
 Keeps true shape of physical features.
 Correct angular relationship.
 Latitude scale is used as distance scale.

3.5.2 The limitations of Mercator charts:

 Cannot be used for Polar Regions.

 Great circles other than meridians and the equator appear curved.
 Unequal area representation.
 Straight lines are not the shortest distances except the equator and meridians.
 Distance scale changes with the change of latitude.

3.5.3 Meridional parts (M.P):

The meridional parts of any latitude are the number of longitude units in the length of a
meridian between the parallel of that latitude and the equator. This unit is used to
construct a Mercator chart as the distance between two consecutive latitudes equals 60
meridional parts at any part of the chart. Since the latitude and distance scale at any part

49
of the chart is proportional to the secant of the latitude of that part, therefore; D.M.P
between any two consecutive latitudes is proportional to the secant of the latitude at that
part, this scale continually increases as it recedes from the equator until it is infinity when
it reaches the pole.

To find the meridional parts (M.P) of any latitude:

The number of meridional parts of any latitude is tabulated in Norie’s tables. To find
difference of meridional parts (D.M.P):
1. F and T are on the same side of the equator, Subtract M.P.T from M.P.F.
2. F and T are on the opposite side of the equator, Add M.P.T to M.P.F.

Fig. (2-19) meridional parts table.

50
3.6 A comparison between Gnomonic projection and Mercator projection:

Mean of comparison Gnomonic projection Mercator projection

Great circles Appear as straight lines. Appear curved.

Rhumb line Appears curved. Appears as a straight line.

Meridians Straight lines converging towards Straight lines parallel to

the nearer pole. each other, equidistant and
perpendicular to the
equator.
Bearings The bearing of any point from Correct representation of
the point of tangency is correctly any point on the chart.
represented.
Distortion Distortion increases as the Amount of distortion in any
distance from the point of latitude is governed by the
tangency increases. secant of that latitude.
Use 1. In sailing along a great 1. In rhumb line
circle. sailing.
2. In polar charts. 2. Used for latitudes
3. In charts whose natural less than 70° N or S.
scale is more than 1: 3. In charts whose
50000. natural scale is less
than 1: 50000.

51
 Questions

1. What are the desirable properties of preferable projection?

2. What are the requirements that should be considered when constructing marine
charts?

3. Define the following: Natural scale, longitudinal scale, and then mention the
relationship between these two scales.

4. What is the mathematical relationship that controls the construction of Mercator

charts?

5. Compare between mercator and gnomonic charts :

 Mention the use of each type in marine navigation.
 Mention the distinctive features of each type.
 Mention the limitations of each type of these charts.

52
 Chapter Four
Position Lines

4.1 Introduction

In coastal navigation, a position line is the line on the earth's surface that represents the
direction or range between the vessel and any terrestrial object. Hence, a Position line is
the line on which the ship lays in a certain point at a certain moment. However, with one
position line, it is impossible to determine the ships position, as the location of the ship
on that position line is unverified. Therefore, at least two position lines are required to
intersect with each other and allocate the ship's position. Position lines are classified into
two main types, straight position line (bearing) and circular position lines which mean
(range) those will be our concern in this chapter.

4.2 Straight position line

A bearing, or a line of bearing, is the direction of one terrestrial point from the vessel
expressed as an angle, measured from a reference 360° in a clockwise direction. There
are several references from which bearings are stated.
1. Bearings which are stated with reference to true north are called true bearings.
2. Bearings which are stated with reference to gyro north are called gyro bearings (i.e.
bearings observed using the gyro compass).
3. Bearings which are stated with reference to compass north are called compass
bearings (i.e. bearings observed using the magnetic compass).
4. Another way of stating bearings is with reference to the fore and aft line of the ship,
such bearing is known as relative bearings.

In addition, another important type of bearing is the transit bearing. Transit bearing is
considered an important type of straight position lines as the compass error (whether gyro
compass error or magnetic compass error) can be checked through it.

Note: The magnetic compass and the gyro compass have an error that makes the zero
of the compass card indicates another reference rather than the true north. This
is named the gyro north (in case of the gyro compass) or the compass north (in
case of the magnetic compass). However, as a matter of fact, any bearing to be
plotted on the chart must be corrected to true bearing, as the reference on the
nautical chart is the true north. Thus, the bearing observed from the vessel to
the terrestrial object using the gyro or the magnetic compass must be corrected
to true bearing before being plotted on the nautical chart i.e. its reference must
be the true north. Currently, let us discuss the above types of bearing
independently.

4.2.1 True bearings

True bearing is the bearing that is measured relative to the true north and is plotted
directly on the chart using the printed compass rose.

53
4.2.2 Gyro bearing

Is the visual bearing or radar bearing (if the radar is gyro stabilized), obtained from the
vessel to a specified object such lighthouse, light buoy or a light vessel using the gyro
compass. At sea, onboard vessels, the visual bearing is obtained by placing the azimuth
circle over the top of the repeater of the gyro compass, and by looking to the object
through the front peep vane and the far vertical wire in the sight van, and reading the
bearing of the object on the compass card through the reflection mirror as shown in the
figure below, this bearing is called Gyro Bearing.

The azimuth circle The azimuth circle placed

over the gyro repeater
4.2.2.1 Gyro error

The gyro compass, as mentioned before, has an error. The gyro error is the horizontal
angle between the gyro north and the true north. This error results from the fact that
although the gyroscopic compass can be extremely accurate, it seldom points exactly to
the true north. The error can be of a high or low value. The high error means that the
gyro reading (which is referred to the gyro north) is greater than its equivalent true
bearing (which is referred to the true north) and visa versa. Thus, before plotting the gyro
bearing on the chart, it must be corrected to true bearing as shown below:
 To change from gyro bearing to true bearing, subtract the gyro error if it is of a high
value and add the gyro error if it is of low value.
 To change from true bearing to gyro bearing, add the gyro error if it is of a high value
and subtract the gyro error if it is of low value.

Example 1

If the gyro bearing of Cap Griz Nez lighthouse is 052°(G) and the gyro error is 2°high
find the true bearing and plot it on the chart.
Solution

Gyro bearing 052°

Gyro error 002° H

True bearing = 050° T

54
Example 2

If the gyro bearing of Cap de Alpreach is 047° (G) and the gyro error is 3° (Low) find the
true bearing and plot it on the chart.
Solution

Gyro bearing 047° G

Gyro error 003° L

True bearing = 050° T

Example 3

If the true bearing of Varne light vessel is 145° (T) and the gyro error is 3° (low) find the
gyro bearing to Varne light vessel.

Solution

True bearing 145° T

Gyro error 003° L

Gyro bearing = 142° G

Example 4

If the true bearing of Varne light vessel is 320° (T) and the gyro error is 2° (high) find the
gyro bearing to Varne light vessel.

Solution

True bearing 320° T

Gyro error 002° H

Gyro bearing = 322° G

Note: the concept is applied to the gyro course and true course.

It is well cleared from what is stated above that the gyro bearing which is one of the
straight position lines can be easily converted to true bearing for the purpose of plotting it
on the chart. Although, there is another type of bearing that the officer of the watch can
use in plotting the ship's position using the magnetic compass and that is the magnetic
compass bearing or compass bearing (as referred to onboard ships).

55
4.2.3 Compass bearing

Is the visual bearing obtained from the vessel to a specified object such lighthouse, light
buoy or a light vessel using the magnetic compass. At sea, onboard vessels, the visual
compass bearing is obtained by placing the azimuth circle over the top of the magnetic
compass, and by looking to the object through the front peep vane and the far vertical
wire in the sight van, and reading the bearing of the object on the compass card through
the reflection mirror as shown in the figure below, this bearing is called compass Bearing.

Similar to the gyro compass, the magnetic compass has an error named the total error.
The total compass error consists of two factors the variation and deviation, that when
known, the total error of the magnets compass can be calculated hence compass bearing
can then be converted to true bearing. Due to the variation and deviation, the needle of
the magnetic compass card will not indicate the true north; it will indicate another
reference called the compass north.

4.2.3.1 Variation

Due to the fact that the true and magnetic poles do not coincide, an angle is formed
between the true north and the magnetic north called variation. This means that, the
needle of the magnetic compass card will not indicate the true north; it will indicate the
magnetic north with an angle named the variation. Variation changes with the change of
position and is named west if the magnetic north lies to the left of the true north and east
if the magnetic north lies to the right.

56
 True True
Magnetic North North Magnetic
North North

W Var E Var

West variation East variation

Variation is found printed on the marine charts for a certain year together with a note of
the annual change. This information changes from chart to chart according to the place
covered by the chart and the year of calculating the variation in that specific place.
However, the navigator must always apply the annual change in his/her calculation to
find out the variation at the present year. In addition, variation may also be obtained from
special isogonics charts on which all places of equal variation are joined by isogonic lines
and known as isogonals.

How to calculate the variation?

The value of the variation is labeled on the compass rose printed on the charts although
this value belongs to a certain year. Therefore, it must be changed to the present year
before being used. Here is an example to illustrate what is mentioned above:

Example:

90 00’ W 1991 (8’E)

Calculate the Var. for 2012

Solution:
1- 2012 - 1991 = 21 year
2- 8’ x 21 = 168 min
3- 168 / 60 = 20 48’ (Rate of change)

4- 90 00' W
20 48' (Rate of change)
0
6 12' W = 6.2 W

VAR. FOR 2012 = 60 12 W = 6.2 W 57

NOTE: The variation is written on the compass rose in two ways:
a) 90 00’ W (1991) (8’ E)
(8' E) = annual change to the east.
OR
b) 90 00’ W (1991) decreasing about 8’ annually (does not exist any more).

Example 1

If the variation on the chart is 050 55’ W (1979), decreasing about 4’ annually. Find the
variation for year 2002 & 2007.

Solution:

2002 - 1979= 23 year 2007 – 1979 = 28 year

23 ×4 min = 92 min (Rate of change) 28 ×4 min = 112 min (Rate of change)
92 ÷ 60 = 01° 32’ (Rate of change) 112’ ÷ 60 = 01° 52’ (Rate of change)
05° 55’ W 05° 55’ W
01° 32 01° 52’
04° 23 W = 4.38 W 04° 03’ W = 4.05W

Note: When subtracting, if the rate of change is greater than the variation on the
chart then you must change the sign of the result.

Example 2

If the variation on the chart is 04° 00' E (1960), decreasing about 6’ annually. Find the
variation for year 2007.

Solution:

2007 - 1960 = 47 year

47 ×6 min = 282 min (Rate of change)
282’ ÷ 60 = 04° 42’ (Rate of change)
040 00 E
040 42

000 42 W = 0.7 W

Example 3

If the variation on the chart is 04° 00' E (1960) (2' E). Find the variation for year 2007.

58
Solution:

2007 - 1960 = 47 year

47 × 2 min = 94 min (Rate of change)
94’ ÷ 60 = 01° 43’ (Rate of change)

040 00 E
010 43 E
050 43 E = 5.7 W

4.2.3.2 Deviation

With a magnetic compass onboard a ship, the needle of the magnetic compass card will
not indicate the magnetic north. Due to the ship's magnetism, the needle will indicate
another reference called the compass north, and the angle between the magnetic north and
the compass north will be called the deviation. Deviation changes with change of the
ship's compass course and it is named east if the compass north lies to the right of the
magnetic north and west if it lies to the left.

Magnetic Magnetic
North North
Compass Compass
North North

W dev
E dev

West deviation East deviation

How to calculate the deviation

The deviation is calculated using the ship's deviation table. Ship's deviation table
represents the ship's compass course and the deviation corresponding to each compass
course (ship's compass course is normally in the deviation table at intervals of ten
degrees). Deviation can also be represented in a graphical form on a deviation curve.

How to use a deviation table

On a deviation table tabulated at ten degree intervals, the deviation for a compass course
010° C or 020° C for instance can be read directly from the table. However, for an
intermediate compass course, like for instance 024° C it is necessary to interpolate

59
between the two deviations tabulated for the compass courses which lies either sides of
024° C as shown below:

Example 1

Find the deviation corresponding to compass course 024° C from the attached deviation
table. Compass course Deviation
010° 2.6 E
020° 1.8 E
030° 1.0 E

Solution

Since the compass course 024° C lies between the compass courses 020° C and 030° C,
therefore an interpolation must be done as follows:

020° C 1.8 E
4° x
10° 024° C Y 0.8

030° C 1.0 E
10° 0.8
4° x

X = (4 x 0.8) ÷ 10
X = 0.32

X is the rate of change of deviation from the deviation corresponding to compass course
020° C (1.8 E) to the unknown deviation corresponding to compass course 024° C (Y).
Since the interpolation is taken with compass course 020° C and the deviation is
decreasing downwards, therefore, the numerical value of x (0.32) will be subtracted from
the deviation corresponding to compass course 020° C:
Deviation corresponding to compass course 024° C = 1.8 – 0.32 = 1.48° E

4.2.3.3 Total compass error

Hence, it can be said that, the angle between the true north and the magnetic north is
named variation and the angle between the magnetic north and the compass north is
named deviation. Consequently, by knowing the value of the variation and the deviation,
the total error of the magnetic compass is known. In other words, compass error can be
defined as the horizontal angle between the true north and the compass north. It is named
east if the compass north lies to the right of the true north and west if it lies to the left.

60
 True True
Compass North North Compass
North North

West error East error

How to calculate the total compass error

For the purpose of this calculation, the east direction will take a positive sign (+) and the
west direction will take a negative sign (-). The total compass error is a combination
between the variation and deviation and the following formula can be used in the
calculation:

Total error = Variation ± Deviation

+ sign when the variation and deviation have the same name (i.e. both are east or both are
west). In this case, the total error will be the algebraic sum of the variation plus the
deviation and the sign of the total error will be either east or west.

− sign when the variation and deviation have different name (i.e. on of them east and the
other is west). In this case, the total error will be the difference between the variation and
the deviation and the sign of the total error will either east or west depending on the sign
of the bigger direction.

Example 1

If the variation is 4.0° E and the deviation is 2.0° E. Find the total compass error?

Solution

Total error = variation ± deviation

Total error = 4.0 +2.0
Total error = 6.0° E

Example 2

If the variation is 6.0° E and the deviation is 4.0° W. Find the total compass error?

61
Solution

Total error = variation ± deviation

Total error = 6.0 − 4.0
Total error = 2.0° E

Example 3

If the variation is 2.0° E and the deviation is 3.0° W. Find the total compass error?

Solution

Total error = variation ± deviation

Total error = 2.0 − 3.0
Total error = 1.0° W

However, after calculating the total compass error, the compass bearing can be changed
to true bearing by applying the following formula.

C AD E T

To true
From compass

Add the
error If the error
is east

Example 1

If the compass bearing is 050° C and the total error is 2.0° E. Find the true bearing?

Solution

Compass bearing = 050° C

Total error = 2.0° E

True bearing = 052° T

62
 C AD E T

052° T
050° C

Add the
error 2.0° E

Note: if the total error is west the formula will be inversed i.e. the total error will be
subtracted from the compass bearing to find the true bearing.

Example 1

If the compass bearing is 242° C and the total error is 2.0° W. Find the true bearing?

Solution

Compass bearing = 242° C

Total error = 2.0° W

True bearing = 240° T

Note: the same concept is applied to the compass course.

4.2.4 Transit bearing

When two objects are seen exactly in line with one another they are said to be in transit.
Hence, at sea, transit bearing can be defined as the extension of the line joining two
conspicuous charted objects in sight of the vessel. When the chosen charted objects are
very close together it is difficult to judge if they are exactly in line. In general, the greater
the horizontal distance between the two objects the greater is the accuracy of the position
line they give.

It should be noted from the previous two types of position line (gyro bearing and compass
bearing) that the accuracy of position line depends on an accurate knowledge of the
compass error (gyro or magnetic compass), which must be applied to the gyro bearings or
the compass bearings to obtain true bearings. In the case of the transit bearing, this does
not apply.

One of the officers of the watch duty is to recognize the error in the gyro compass and the
magnetic compass. The transit bearing is very effect in this case. The two objects used in
the transit bearing are recognized well on the chart. The transit line drawn on the chart
between the two objects is a true transit bearing and the same transit bearing is obtained
by the navigator using the gyro compass or the magnetic compass. Therefore, by knowing

63
the difference between the true transit bearing (which is drawn on the chart) and the gyro
transit bearing or compass transit bearing, the error of the compass (whether gyro or
magnetic compass) can be easily recognized as shown in the example:

Example 1

If the compass transit bearing of lighthouse A and B is 285° C. Find the total compass
error if the true transit bearing measured on the chart is 290° T.

Solution

1. On the chart, join the two lighthouses with a line and extend it beyond the two
lighthouses in the direction of the vessel. This is the true transit bearing.

2. Shift the true transit bearing to the compass rose take the reading.

3. Compare between the reading of the true transit bearing and the compass transit
bearing. This will be the total compass error.

Compass transit bearing = 285° C

True transit bearing = 290° T

Total compass error = 5° E

Note: in the transit case, the required total compass error is named east or west
according to the following statement:
Compass best error west
Compass least error east

True transit bearing

64
4.2.4.1 Deviation from transit bearing

In case of a compass transit bearing, the deviation can be calculated without a deviation
table. The total compass error is known from the difference between the compass transit
bearing and the true transit bearing and the variation is known from the compass rose
(after being corrected to the present year). Therefore, there is one unknown value in the
following equation which can be easily known (bear in mind that the east takes a positive
sign and the west takes a negative sign):

Total error = variation + Deviation

Using the previous example with a variation 2.0 E, calculate the deviation.

Solution

Total compass error = 5.0° W and the variation = 2.0° E

Applying the above equation algebraically:
−5.0 = 2.0 + deviation
Deviation = − 5.0 − 2.0 = − 7.0 (and (−) sign means west)
Therefore, deviation = 7.0° W

Using the previous example with a variation 2.0 W, calculate the deviation.

Solution

Total compass error = 5.0° W and the variation = 6.0° W

Applying the above equation algebraically:
−5.0 = − 6.0 + deviation
Deviation = − 5.0 + 6.0 = + 1.0 (and (+) sign means east)
Therefore, deviation = 1.0° E

Example 2

If the gyro transit bearing for Beachy Head & Royal Sovereign is 062° G. Find the gyro
error if the true transit bearing measured on the chart is 060° T.

Solution

Gyro transit bearing = 062° G

True transit bearing = 060° T

Gyro error = 2° H

Note: in this case, the required gyro error is named high or low according to the
gyro reading (if the gyro reading is higher than the true reading, the error is
high and visa versa)

65
4.2.5 Relative bearing

As it is mentioned earlier in this chapter, the true bearing have been stated with reference
to the true north and the gyro bearing and compass bearing with reference to the gyro
north and compass north respectively. Another way of stating bearing is with reference to
the ships fore and aft line such bearing is known as relative bearing. Relative bearing can
be defined as the angle between the ships’s heading (fore-and-aft line) or the true course
to steer and the line of sight joining the ship and the object (i.e. it is the bearing measured
relative to the ship’s heading).

True north
True course
True
bearing
Relative
bearing Object

However, there are two ways for measuring relative bearing. Relative bearing can be
stated in 360° notation (0° to 360° clockwise) or from 0° to 180° on the starboard side
(green), 0° to 180° on the port side (red).

Example

If the true course to steer = 170° T and the relative bearing of Bill of Portland lighthouse
= 030°. Find the true bearing of Bill of Portland lighthouse.

Solution

In this case, the relative bearing is measured clockwise from the ships heading.
Therefore:
True bearing = true course + relative bearing
170° + 030°
True bearing = 200° T

66
Example

If the true course to steer = 030° T and the relative bearing of Dungeness lighthouse =
070° green. Find the true bearing of Dungeness lighthouse.

Solution

In this case, the relative bearing is measured from 0° to 180° on the starboard side of the
ships heading (070° green)
Therefore:
True bearing = true course ± relative bearing
030° + 070°
True bearing = 100° T

Note: in this case, the relative bearing is added to the true course to steer because the
lighthouse is on the starboard side of the ship's bow (ship's heading).

Example

If the true course to steer = 030° T and the relative bearing of Dungeness lighthouse =
020° Red. Find the true bearing of Dungeness lighthouse.

Solution

In this case, the relative bearing is measured from 0° to 180° on the port side of the ships
heading (020° red)
Therefore:
True bearing = true course ± relative bearing
030° − 020°
True bearing = 010° T

Note: in this case, the relative bearing is subtracted from the true course to steer because
the lighthouse is on the port side of the ship's bow (ship's heading). In addition, in case of
a red relative bearing (on the port side) and it is greater than the true course, add 360° to
the true course first then subtract from it the relative bearing.

Example

If the true course to steer = 030° T and the relative bearing of Dungeness lighthouse =
040° Red. Find the true bearing of Dungeness lighthouse.

Solution

In this case, the relative bearing is measured from 0° to 180° on the port side of the ships
heading (040° red) and is greater than the true course.

67
Therefore:
True bearing = true course ± relative bearing
(030° + 360) − 040°
True bearing = 350° T
In addition, relative bearing can also be measured from right ahead e.g. 40° on the
starboard bow, 20° on port bow. They can also be measured from right astern e.g. 25° on
the port quarter, 35° on the starboard quarter. Another important relative bearing is the
beam bearing.

4.2.6 Beam bearing

Is the bearing which lies at right angle to the course steered or the ships heading. This
means that, the beam bearing always makes a 90° angle from those references.

4.3 Circular position lines (distance off)

The circular position line is the distance off taken from the ship to any object such as
(lighthouse, light buoy, light vessel, land mark….). These objects are shown on the charts
with a special symbols and special characteristics. The distance off, is represented by an
arc on which the vessel lies on it in a certain point at a certain moment. Remember, the
ship's position can not be determined by one position line, at least two are needed.
However, there are two types of circular position lines:
1. Direct distance (Distance by Radar)
2. Indirect distance such as:
2.1 Vertical sextant angle
2.2 Horizontal sextant angle
2.3 Light just appear or disappear

4.3.1 Direct distance (Distance by radar)

It is obtained by measuring the distance off between the vessel and the target using the
radar and plotting it directly on the chart using the compass. Remember that, the distance
is measured using the part of the latitude scale opposite to the navigating area.

4.3.2 Indirect Distance

Is the distance that is obtained indirectly using one of the following three ways:
1. Vertical sextant angle
2. Horizontal sextant angle
3. Lighthouse first appear or disappear

68
4.3.2.1 Distance by Vertical Sextant Angle

The vertical sextant angle is the angle measured from the vessel between the top of the
light house and its bottom. This angle is measured by an instrument called the sextant and
is converted to distance.

This distance is considered an indirect distance and it can be obtained by three ways:
 Distance = (1.855 × height of light house) ÷ vertical sextant angle in minutes.
 Distance = height of lighthouse ÷ tan vertical sextant angle.
 Using the Nories table (section: distance by vertical sextant angle).

Notes:
The sextant instrument has an error named the index error and the numerical value of this
error is always in minutes. Hence, the vertical sextant angle in the first way must be
corrected for such an error to insure a correct distance calculation. The index error could
be on arc error or off arc error. In case of an on arc error, it is subtracted from the sextant
angle and in case of an off arc error it is added to the sextant angle.

Example 1

If the V.S.A of Dungeness lighthouse is 01° 12'.0 and the index error is 1'.2 on arc. Find
the distance off lighthouse.

Solution

Since the sextant has an inex error (1'.2 on arc), the V.S.A must be correct to C.V.S.A:

V.S.A = 01° 12'.0

I.E = 1'.2 on arc −

C.V.S.A = 01° 10'.8

69
C.V.S.A in min = 70'.8

The height of Dungeness lighthouse = 40 m

Distance off = (1.855 × height of lighthouse) ÷ C.V.S.A in min

Distance off = (1.855 × 40) ÷ 70'.8
Distance off = 1.0 nautical miles

Example 2

If the V.S.A of Dungeness lighthouse is 00° 50'.0 and the index error is 1'.2 off arc. Find
the distance off lighthouse.

Solution

Since the sextant has an inex error (1'.2 on arc), the V.S.A must be correct to C.V.S.A:

V.S.A = 00° 50'.0

I.E = 1'.2 off arc +

C.V.S.A = 00° 51'.2

C.V.S.A in min = 51'.2

The height of Dungeness lighthouse = 40 m

Distance off = (1.855 × height of lighthouse) ÷ C.V.S.A in min

Distance off = (1.855 × 40) ÷ 51'.2
Distance off = 1.4 nautical miles.

4.3.2.3 Lighthouse just Appear or disappear.

To have a circular position line (distance off) from a lighthouse, the types of light ranges
of lighthouses must be discussed fist:
There are three types of light range of a lighthouse named:
A. Geographical range.
B. Nominal range.
C. Luminous range.

A. Geographical Range

Geographical range is (theoretically) the maximum distance for the observer to see the
light. This range depends on the height of light and the height of eye of the observer. (i.e.
it depend on the earth’s curvature).

70
 (d1 + d2 = geographic Range)

The geographical range can be obtained by one of the following ways:

Norise table:
a- From the section (Distance of sea horizon).OR
b- From the section (Extreme range table)
Admiralty List of light tables (From the Geographical range table).
Horizon equation:

Norise table

It can be done by entering in (Distance of sea horizon) table two times. The first time, the
table is entered with the height of eye of the observer and finds d1 (distance 1). The
second time it is entered with the height of the lighthouse and finds d2 (distance 2). By
adding d1 and d2, the value of the geographical range is obtained.

Admiralty List of light tables: From the (Geographical range table).

It is done by entering the geographical range table with the height of lighthouse (column)
and the height of eye of the observer (row). Where the row and column meet, this is the
geographical range of the light house.

Example

Find the geographical range of Ras el tin lt.ho using the geographical range table. Notice
that, the height of eye of the observer is 20 meter and the height of light house is 30 meter

71
Horizon equation:

G.R. = 2.03 √H + 2.03 √h

Where H = height of lighthouse.
h = height of the eye of the observer.

B. Nominal Range

The nominal range is written on the cart in capital M. It is calculated assuming that the
visibility in the area according to the weather phenomena is 10 miles.

C. Luminous Range

Is the maximum distance through which you can sea the light in a certain moment. It
depends on:
 The intensity of light.
 Visibility in the area.

Luminous range is obtained using the luminous range diagram found in the list of light
publications using the (nominal range + visibility).

Example

Find the luminous range of Folkestone lighthouse if the visibility is 5 miles

72
 Solution

 From the chart, near Folkestone lighthouse, the nominal range is writhen in capital M
(22 M)
 From the horizontal axes of the luminous range diagram at the figure 22 (nominal
range) drop down a straight line to intersect the given visibility curve (5 miles).

 At the point of intersection, drawn another straight line to left to intersect the
luminous column.

 Read the numerical value of the luminous range of the lighthouse. Folkestone
lighthouse which is in this example will be 13 miles.

13 miles

73
74