Great Circle Tracks / Convergency

Convergency

• Convergency is how much the great circle track changes between two points
• Convergency = change of longitude x sin mean latitude
• Convergency is 0 at the Equator (i.e. sin 0 = 0) and maximum at the Poles (i.e. sin 90 = 1)

Great circles and rhumb lines

• A great circle is the shortest distance between 2 points
• Rhumb lines are lines of constant direction
• Rhumb line track = average great circle track
• If two points are on the same latitude, rhumb line track is equal to the great circle track at the midpoint
• If two points are on the same latitude, the rhumb line track between them is either 090° or 270°
• Rhumb lines always lie on the Equatorial side of the great circles
• Equator and meridians together with their anti-meridians are both great circles and rhumb lines and are the only great circles that do not change track direction (rhumb line and great circle are exactly the same)

Conversion angle

• The angle between great circle track and rhumb line track is called conversion angle
• Conversion angle = 1/2 convergency

Calculating great circle track for return flight

• If asked to calculate great circle track from B to A (e.g. return flight) , calculate first great circle track on B and then calculate the reciprocal

Learning to calculate great circle tracks without diagrams

• In Northern hemisphere if you move in easterly direction great circle track increases
• In Northern hemisphere if you move in westerly direction great circle track decreases
• In Southern Hemisphere, if you move in easterly direction great circle track decreases
• In Southern Hemisphere if you move in westerly direction great circle track increases

Exercise 1.1

• A 25° N 055° E
• B 55° N 045° W
• What is the convergency between A and B?

Answer

• Change of longitude = 45° + 55° = 100°

To find mean lat add the latitude and then divide by 2:

• 25° + 55° = 80°
• 80° / 2 = 40°

• Convergency = change of long x sin mean lat
• Convergency = 100° x sin 40
• Convergency = 64°

This means that great circle track from position A to B has changed by 64° (decreased by 64° as we are in the Northern hemisphere and move to the West)

Exercise 1.2

• A 45° N 040° W
• B 65° N 020° E
• What is the convergency between A and B?

Answer

• Change of longitude = 40° + 20° = 60°

To find mean lat add the latitude and then divide by 2:

• 45° + 65° = 110°
• 110° / 2 = 55°

• Convergency = change of long x sin mean lat
• Convergency = 60° x sin 55
• Convergency = 49°

This means that great circle track from position A to B has changed by 49° (increased by 49° as we are in the Northern hemisphere and move to the East)

Exercise 1.3

• A 55° S 110° E
• B 75° S 150° E
• What is the convergency between A and B?

Answer

• Change of longitude = 150° – 110° = 40°

To find mean lat add the latitude and then divide by 2:

• 55° + 75° = 130°
• 130°/ 2 = 65°

• Convergency = change of long x sin mean lat
• Convergency = 40° x sin 65
• Convergency = 36°

This means that great circle track from position A to B has changed by 36° (decreased by 36° as we are in the Southern hemisphere and move to the East)

Exercise 1.4

• A 30° S 100° W
• B 40° S 140° W
• What is the convergency between A and B?

Answer

• Change of longitude = 140° – 100° = 40°

To find mean lat add the latitude and then divide by 2:

• 30° + 40° = 70°
• 70° / 2 = 35°

• Convergency = change of long x sin mean lat
• Convergency = 40° x sin 35
• Convergency = 23°

This means that great circle track from position A to B has changed by 23° (increased by 23° as we are in the Southern hemisphere and move to the West)

Exercise 2.1

• A 25° S 035° E
• B 25° S 175° W
• What is the conversion angle?

Answer

To find the conversion angle, we first need to find the convergency and then divide it by 2

• Change of longitude = 35° + 175° = 210°
• 360° – 210° = 150°

• Convergency = change of long x sin mean lat
• Convergency = 150° x sin 25
• Convergency = 63°

• Conversion angle = 1/2 convergency
• Conversion angle = 63° / 2
• Conversion angle = 31.5°

Therefore the angle between great circle track and rhumb line track is 31.5°

Exercise 2.2

• A 65° N 035° W
• B 25° N 115° W
• What is the conversion angle?

Answer

To find the conversion angle, we first need to find the convergency and then divide it by 2

• Change of longitude = 115° – 35° = 80°
• Mean lat = 45°

• Convergency = change of long x sin mean lat
• Convergency = 80° x sin 45
• Convergency = 56°

• Conversion angle = 1/2 convergency
• Conversion angle = 56° / 2
• Conversion angle = 28°

Therefore the angle between great circle track and rhumb line track is 28°

Exercise 3.1

• A 25° N 023° E
• B 75° N 089° E
• Initial great circle track at A is 082° T
• What is the final great circle track at B?

Answer

• Notice that aircraft is in the Northern hemisphere and moves to the East
• This means that great circle track from A to B increases

• Convergency = change of long x sin mean lat
• Convergency = 66° x sin 50°
• Convergency = 50°

As great circle track from A to B increases, final great circle track at B will be 082° + 50° = 132° T

Exercise 3.2

• A 55° N 085° E
• B 25° N 005° E
• Initial great circle track at A is 295° T
• What is the final great circle track at B?

Answer

• Notice that aircraft is in the Northern hemisphere and moves to the West
• This means that great circle track from A to B decreases

• Convergency = change of long x sin mean lat
• Convergency = 80° x sin 40
• Convergency = 51°

As great circle track from A to B decreases, final great circle track at B will be 295° – 51° = 244° T

Exercise 3.3

• A 75° S 070° W
• B 65° S 090° W
• Initial great circle track at A is 250° T
• What is the final great circle track at B?

Answer

• Notice that aircraft is in the Southern hemisphere and moves to the West
• This means that great circle track from A to B increases

• Convergency = change of long x sin mean lat
• Convergency = 20° x sin 70
• Convergency = 18°

As great circle track from A to B increases , final great circle track at B will be 250° + 18° = 268° T

Exercise 3.4

• A 25° S 150° W
• B 15° S 080° W
• Initial great circle track at A is 110° T
• What is the final great circle track at B?

Answer

• Notice that aircraft is in the Southern hemisphere and moves to the East
• This means that great circle track from A to B decreases

• Convergency = change of long x sin mean lat
• Convergency = 70° x sin 20
• Convergency = 24°

As great circle track from A to B decreases, final great circle track at B will be 110° – 24° = 086° T

Exercise 4.1

• A 50° S 150° W
• B 10° S 170° E
• Final great circle track at B is 300° T
• What is the initial great circle track at A?

Answer

• Notice that aircraft is in the Southern hemisphere and moves to the West (passing through Greenwich anti-meridian)
• This means that great circle track from A to B increases

• Change of longitude = 40° (150° + 170° = 320° , 360° – 320° = 40°)
• Convergency = change of long x sin mean lat
• Convergency = 40° x sin 30
• Convergency = 20°

As great circle track from A to B increases, initial great circle track at A will be 300° – 20° = 280° T

Exercise 4.2

• A 65° S 160° E
• B 35° S 120° W
• Final great circle track at B is 070° T
• What is the initial great circle track at A?

Answer

• Notice that aircraft is in the Southern hemisphere and moves to the East (passing through Greenwich anti-meridian)
• This means that great circle track from A to B decreases

• Change of longitude = 80° (160° + 120° = 280° , 360° – 280° = 80°)
• Convergency = change of long x sin mean lat
• Convergency = 80° x sin 50
• Convergency = 61°

As great circle track from A to B decreases, initial great circle track at A will be 070° + 61° = 131° T

Exercise 4.3

• A 55° N 130° E
• B 05° N 130° W
• Final great circle track at B is 120° T
• What is the initial great circle track at A?

Answer

• Notice that aircraft is in the Northern hemisphere and moves to the East (passing through Greenwich anti-meridian)
• This means that great circle track from A to B increases

• Change of longitude = 100° (130° + 130° = 260° , 360° – 260° = 100°)
• Convergency = change of long x sin mean lat
• Convergency = 100° x sin 30
• Convergency = 50°

As great circle track from A to B increases, initial great circle track at A will be 120° – 50° = 070° T

Exercise 4.4

• A 70° N 150° W
• B 20° N 150° E
• Final great circle track at B is 220° T
• What is the initial great circle track at A?

Answer

• Notice that aircraft is in the Northern hemisphere and moves to the West (passing through Greenwich anti-meridian)
• This means that great circle track from A to B decreases

• Change of longitude = 60° (150° + 150° = 300° , 360° – 300° = 60°)
• Convergency = change of long x sin mean lat
• Convergency = 60° x sin 45
• Convergency = 42°

As great circle track from A to B decreases, initial great circle track at A will be 220° + 42° = 262° T

Exercise 5.1

• A 35° S 055° E
• B 55° S 075° E
• Rhumb line track is 080° T
• What is the great circle track at A and B?

Answer

For this exercise, since we know the rhumb line track, we first need to find the conversion angle before we calculate the great circle tracks

• Convergency = change of long x sin mean lat
• Convergency = 20° x sin 45
• Convergency = 14 °

• Conversion angle = 1/2 convergency
• Conversion angle = 7°

We are in the Southern Hemisphere and B is to the East of A, therefore great circle track from A to B decreases

• Great circle Track at A = 080° + 7° = 087° T
• Great circle Track at B = 080° – 7° = 073° T

Exercise 5.2

• A 50° S 150° E
• B 30° S 060° E
• Rhumb line track is 250° T
• What is the great circle track at A and B?

Answer

For this exercise, since we know the rhumb line track, we first need to find the conversion angle before we calculate the great circle tracks

• Convergency = change of long x sin mean lat
• Convergency = 90° x sin 40
• Convergency = 58°

• Conversion angle = 1/2 convergency
• Conversion angle = 29°

We are in the Southern Hemisphere and B is to the West of A, therefore great circle track from A to B increases

• Great circle Track at A = 250° – 29° = 221° T
• Great circle Track at B = 250° + 29° = 279° T

Exercise 6.1

• A 72° S 084° W
• B 24° S 034° W
• Initial great circle track at A is 298° T
• What is the great circle track from B to A?

Answer

Notice that the exercise asks for the great circle track from B to A which is going to be the reciprocal of the final great circle track at B

Notice that aircraft is in the Southern Hemisphere and moves to the East. Therefore great circle track from A to B decreases

• Convergency = change of long x sin mean lat
• Convergency = 50 x sin 48
• Convergency = 37°

As great circle track decreases, final great circle track at B will be 298° – 37° = 261° T

The question however asks for great circle track from B to A, which will be the reciprocal of final great circle track at B: 261° – 180° = 081° T

Exercise 6.2

• A 75° N 055° W
• B 25° N 100° W
• Initial great circle track at A is 285° T
• What is the great circle track from B to A?

Answer

Notice that the exercise asks for the great circle track from B to A which is going to be the reciprocal of the final great circle track at B

Notice that aircraft is in the Northern Hemisphere and moves to the West. Therefore great circle track from A to B decreases

• Convergency = change of long x sin mean lat
• Convergency = 45 x sin 50
• Convergency = 34°

As great circle track decreases, final great circle track at B will be 285° – 34° = 251° T

The question however asks for great circle track from B to A, which will be the reciprocal of final great circle track at B: 251° – 180° = 071° T

Exercise 7.1

• A 78° S 156° W
• B 48° S 149° E
• Initial great circle track at A is 237° T
• What is the rhumb line track from A to B?
• What is the rhumb line track from B to A?

Answer

• Rhumb line track is the same in all positions between A and B, as by definition it’s a line of constant direction
• Rhumb line track is equal to the average great circle track
• To find the average great circle track we need to find the initial (which is given) and final great circle tracks and then divide by 2

• Change of longitude = 55° (156° + 149° = 305° , 360° – 305° = 55°)
• Convergency = change of long x sin mean lat
• Convergency = 55° x sin 63 = 49°

• Notice that we are in the Southern Hemisphere and move to the West (passing through Greenwich anti-meridian) so great circle track from A to B increases
• Therefore final great circle track at B is: 237° + 49° = 286° T

Average great circle track from A to B:

• 237° + 286° = 523°
• 523°/2 = 261.5 ° T

Therefore rhumb line track from A to B is 261.5 ° T

Rhumb line track from B to A is the reciprocal: 261.5° – 180° = 081.5° T

Exercise 7.2

• A 55° N 025° W
• B 35° N 060° W
• Initial great circle track at A is 292° T
• What is the rhumb line track from A to B?
• What is the rhumb line track from B to A?

Answer

• Rhumb line track is the same in all positions between A and B, as by definition it’s a line of constant direction
• Rhumb line track is equal to the average great circle track
• To find the average great circle track we need to find the initial (which is given) and final great circle tracks and then divide by 2

• Convergency = change of long x sin mean lat
• Convergency = 35° x sin 45
• Convergency = 24°

• Notice that we are in the Northern Hemisphere and move to the West so great circle track decreases
• Therefore final great circle track at B is: 292° – 24° = 268° T

Average great circle track from A to B:

• 292° + 268° = 560°
• 560°/2 = 280° T

Therefore rhumb line track from A to B is 280° T

Rhumb line track from B to A is the reciprocal: 280° – 180° = 100° T

Exercise 8.1

• A 45° N 010° E
• B 45° N 030° W
• What is the great circle tracks at A and B?

Answer

• Notice that latitude of A to B is the same
• This means that rhumb line track is either 090° or 270°
• In this case it’s 270° T as B is to the West of A

As points A and B are in the Northern hemisphere and aircraft moves to the West (passing through Greenwich meridian), great circle track from A to B decreases

We know the rhumb line track, therefore to find the great circle tracks we need to find the conversion angle (1/2 convergency)

• Convergency = change of long x sin mean lat
• Convergency = 40° x sin 45
• Convergency = 28°

• Conversion angle = 1/2 convergency
• Conversion angle = 14°

As great circle track from A to B decreases

• Great circle track at A = 270° + 14° = 284° T
• Great circle track at B = 270° – 14° = 256° T

Exercise 8.2

• A 30° S 050° W
• B 30° S 050° E
• What is the great circle tracks at A and B?

Answer

• Notice that latitude of A to B is the same
• This means that rhumb line track is either 090° or 270°
• In this case it’s 090° T as B is to the East of A

As points A and B are in the Southern hemisphere and aircraft moves to the East (passing through Greenwich meridian), great circle track from A to B decreases

We know the rhumb line track, therefore to find the great circle tracks we need to find the conversion angle (1/2 convergency)

• Convergency = change of long x sin mean lat
• Convergency = 100° x sin 30
• Convergency = 50°

• Conversion angle = 1/2 convergency
• Conversion angle = 25°

As great circle track from A to B decreases

• Great circle track at A = 090° + 25° = 115° T
• Great circle track at B = 090° – 25° = 065° T

Exercise 9.1

• A 65° S 055° E
• B 65° S° X°
• Initial great circle track is 104° T
• What is the longitude of B?

Answer

• Notice that position A and B are on the same latitude
• This means that rhumb line track is either 090° or 270°
• In this case great circle track shows aircraft is flying to the East, so rhumb line track is 090° T

• To calculate the longitude of B, we need to find the change of longitude between A and B using the formula of convergency
• We know the latitude and we can find the convergency by calculating first the conversion angle

Conversion angle is the difference between great circle track and rhumb line track

• Conversion angle = 104° – 90°
• Conversion angle = 14°

• Conversion angle = 1/2 convergency
• Convergency = 28°

• Convergency = change of long x sin mean lat
• Change of long = convergency / sin mean lat
• Change of long = 28° / sin 65
• Change of long = 31°

Aircraft is moving to the East so longitude of B = 55° + 31° = 086° E

Exercise 9.2

• A 30° N 110° W
• B 30° N° X°
• Final great circle track is 260° T
• What is the longitude of B?

Answer

• Notice that position A and B are on the same latitude
• This means that rhumb line track is either 090° or 270°
• In this case great circle track shows aircraft is flying to the West, so rhumb line track is 270° T

• To calculate the longitude of B, we need to find the change of longitude between A and B using the formula of convergency
• We know the latitude and we can find the convergency by calculating first the conversion angle

Conversion angle is the difference between great circle track and rhumb line track

• Conversion angle = 270° – 260°
• Conversion angle = 10°

• Conversion angle = 1/2 convergency
• Convergency = 20°

• Convergency = change of long x sin mean lat
• Change of long = convergency / sin mean lat
• Change of long = 20° / sin 30
• Change of long = 40°

Aircraft is moving to the West so longitude of B = 110° + 40° = 150° W

Exercise 10.1

• A 60° S X°
• B 60° S 120° E
• Initial great circle track is 240° T
• What is the longitude of A?

Answer

• Notice that position A and B are on the same latitude
• This means that rhumb line track is either 090° or 270°
• In this case great circle track shows aircraft is flying to the West, so rhumb line track is 270° T

• To calculate the longitude of A, we need to find the change of longitude between A and B using the formula of convergency
• We know the latitude and we can find the convergency by calculating first the conversion angle

Conversion angle is the difference between great circle track and rhumb line track

• Conversion angle = 270° – 240°
• Conversion angle = 30°

• Conversion angle = 1/2 convergency
• Convergency = 60°

• Convergency = change of long x sin mean lat
• Change of long = convergency / sin mean lat
• Change of long = 60° / sin 60
• Change of long = 69°

• As aircraft moves to the West from A to B, it means that A is to the East of B, therefore longitude of A is 120° + 69° = 189°
• However, longitude can’t be more than 180°
• If the sum of longitude angles is greater than 180°, it means that aircraft has moved to the opposite hemisphere. To calculate the longitude of A we need to subtract this number from 360°
• 360° – 189° = 171° W (passing through Greenwich anti-meridian)

Exercise 10.2

• A 70° N X°
• B 70° N 010° E
• Final great circle track is 110° T
• What is the longitude of A?

Answer

• Notice that position A and B are on the same latitude
• This means that rhumb line track is either 090° or 270°
• In this case great circle track shows aircraft is flying to the East, so rhumb line track is 090° T

• To calculate the longitude of A, we need to find the change of longitude between A and B using the formula of convergency
• We know the latitude and we can find the convergency by calculating first the conversion angle

Conversion angle is the difference between great circle track and rhumb line track

• Conversion angle = 110° – 090°
• Conversion angle = 20°

• Conversion angle = 1/2 convergency
• Convergency = 40°

• Convergency = change of long x sin mean lat
• Change of long = convergency / sin mean lat
• Change of long = 40° / sin 70
• Change of long = 42°

• As aircraft moves to the East from A to B, it means that A is to the West of B, therefore longitude of A is 10° – 42° = – 32°
• -32° means that the aircraft has moved to the opposite hemisphere (passing through Greenwich meridian)
• Therefore longitude of A is 032° W