Convergency, conversion Angle and Departure
Convergency, conversion angle and departure are three of the most calculation-heavy ideas in General Navigation.
They all come from one fact: meridians are parallel at the equator but meet at the poles. Once you can picture that, the formulas stop being things to memorise and start making sense.
Here are the three formulas you need. The rest of this article explains why they work and how to apply them without mixing up the signs.
Quantity | What it measures | Formula |
Departure | East - west distance along a parallel of latitude | Departure (NM) = ch.long (minutes) × cos lat |
Earth convergency | The angle between two meridians, which equals the change in great-circle track between them | Convergency = ch.long × sin (mean) lat |
Conversion angle (CA) | The angle between the great circle and the rhumb line at either end | CA = ½ × convergency |
Ch.long means change of longitude (some books write d.long). All three topics appear under "Basics of Navigation" in the DGCA CPL Air Navigation syllabus, which lists the great circle, small circle, rhumb line, convergency and conversion angle together.
The building blocks: great circles, rhumb lines and ch.long
Before the formulas, keep three definitions clear.
A great circle is any circle on the Earth's surface whose plane passes through the Earth's centre. The shorter arc of a great circle is the shortest distance between two points. A great-circle route crosses each meridian at a different angle.
A rhumb line crosses every meridian at the same angle. That makes it easy to fly on a constant track, but it is usually longer than the great circle.
Meridians are halves of great circles. The equator is a great circle. Every other parallel of latitude is a small circle.
That last point is where departure comes from.
Departure: east-west distance along a parallel
Why departure uses cos lat
One minute of arc along a great circle equals one nautical mile. At the equator, which is a great circle, 1′ of ch.long is therefore 1 NM.
Away from the equator, a parallel of latitude is a smaller circle. Its radius is R cos φ, where R is the Earth's radius and φ is the latitude. The same ch.long covers a shorter arc, shrinking by exactly the factor cos φ.
Departure (NM) = ch.long (minutes) × cos lat
Rearranged: ch.long (minutes) = departure ÷ cos lat
Two quick checks follow straight from this:
At the equator, cos 0° = 1, so departure equals ch.long in minutes.
At the pole, cos 90° = 0. All meridians meet, so there is no east–west distance left.
Worked example 1: departure along 23°N
Question: An aircraft flies due east along 23°N from 072°E to 088°E. What distance does it cover?
Ch.long = 088° − 072° = 16° = 960′
cos 23° = 0.9205
Departure = 960 × 0.9205 = 883.7 NM
Sense check: the answer is less than 960 NM, as it must be anywhere off the equator.
Worked example 2: finding the new longitude
Question: An aircraft at 60°N 005°W flies 300 NM due east. What is its new position?
Ch.long = departure ÷ cos lat = 300 ÷ cos 60° = 300 ÷ 0.5 = 600′ = 10°
Moving east from 005°W by 10° gives 60°N 005°E.
This reverse form is a common way to test departure. The trap is going the wrong way across the prime meridian or the 180° meridian. Always sketch the longitudes before adding or subtracting.
Note: Flying along a parallel is a rhumb-line route, not the shortest route. The great circle between two points on the same parallel (other than the equator) is shorter, because it bulges towards the pole.
Earth convergency: the angle between meridians
What convergency is
Earth convergency is the angle between two meridians at a given latitude. Think of it as how much two "north" directions point towards each other.
At the equator, the meridians are parallel, so convergency is 0°.
At the pole, they meet at an angle equal to the full ch.long, so convergency equals ch.long.
In between, convergency changes with the sine of the latitude.
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Convergency = ch.long × sin lat
For two points at different latitudes, use the mean latitude.
Convergency is the change in great-circle track
This is the idea that makes convergency useful. A great circle is a "straight line" on the globe. Each meridian it crosses points north in a slightly different direction, so the track angle measured from each meridian must change. That change equals the angle between the meridians, which is the convergency.
So, for a great-circle route from A to B:
GC track at B = GC track at A ± convergency
Whether you add or subtract depends on hemisphere and direction, covered in the next section.
How fast convergency grows with latitude
Because convergency follows a sine curve, it rises quickly at low latitudes and levels off towards the pole. For a 10° ch.long, convergency is 5° at 30°, 7.07° at 45°, 8.66° at 60° and the full 10° at the pole.
Worked example 3: convergency on the same parallel
Question: Find the earth convergency between 20°N 070°E and 20°N 090°E.
Ch.long = 20°
Convergency = 20 × sin 20° = 20 × 0.3420 = 6.84° ≈ 6.8°
We use this result in the next section.
Conversion angle: great circle versus rhumb line
Where the great circle lies
Between any two points not on the same meridian or on the equator, the great circle and the rhumb line are different routes. The rule that settles every sign question is:
The great circle always lies on the polar side of the rhumb line.
The conversion angle (CA) is the angle between the two at either end. To a close approximation:
CA = ½ × convergency
The rhumb-line track sits roughly halfway between the great-circle track at the start and the great-circle track at the end.
Continuing worked example 3: from 20°N 070°E to 20°N 090°E, the rhumb line runs along the parallel at 090°(T). Convergency is 6.8°, so CA = 3.4°.
The great circle starts poleward (north of east): 090 − 3.4 = 086.6°(T)
It finishes equatorward (south of east): 090 + 3.4 = 093.4°(T)
Check: 093.4 − 086.6 = 6.8°, which is the convergency.
The hemisphere and direction rules
Applying "great circle on the polar side" gives four cases:
Hemisphere | Direction | GC track at start | GC track along the route |
Northern | Eastbound | RL − CA | Increases |
Northern | Westbound | RL + CA | Decreases |
Southern | Eastbound | RL + CA | Decreases |
Southern | Westbound | RL − CA | Increases |
You don't need to memorise the table. Sketch the route, put the great circle on the pole side, and read off whether the starting track is clockwise or anticlockwise of the rhumb line. Another way to say it: the great circle starts poleward and finishes equatorward.
Worked example 4: finding the hemisphere and latitude
Question: Two points are on the same parallel with a ch.long of 14°. The rhumb-line track from P to Q is 090°(T), and the great-circle track at P is 094°(T). Which hemisphere are they in, and at what latitude?
The great circle starts at 094°, which is south of east. It must therefore bulge towards the South Pole, so the points are in the southern hemisphere.
CA = 094 − 090 = 4°, so convergency = 2 × 4 = 8°
sin lat = convergency ÷ ch.long = 8 ÷ 14 = 0.5714, so lat = 34.8°S (approximately)
Worked example 5: points at different latitudes
Question: From A (40°N 010°W) to B (50°N 010°E), the initial great-circle track at A is 048°(T). Find the rhumb-line track A→B and the great-circle track on arrival at B.
Ch.long = 010°W to 010°E = 20°
Mean latitude = (40 + 50) ÷ 2 = 45°N
Convergency = 20 × sin 45° = 14.1°
CA = 14.1 ÷ 2 = 7.1°
Northern hemisphere, eastbound: the great-circle track increases, and the rhumb line is clockwise of the initial great-circle track.
Rhumb-line track = 048 + 7.1 = 055.1° ≈ 055°(T)
GC track at B = 048 + 14.1 = 062.1° ≈ 062°(T)
How accurate are the exam formulas?
For the angle between two meridians at one specific latitude, convergency = ch.long × sin lat is exact. Using the mean latitude for two different latitudes, and taking CA as exactly half the convergency, are approximations. They are very good over normal distances.
We checked our worked examples against full spherical trigonometry:
Example | Formula answer | Spherical calculation |
3: GC tracks, 20°N 070°E → 090°E | 086.6° → 093.4° | 086.55° → 093.45° |
5: rhumb line, 40°N 010°W → 50°N 010°E | 055.1° | 054.6° |
5: GC track at B | 062.1° | 062.2° |
Use the formulas as taught. Just don't be surprised if a very long route gives a slightly different answer in a flight-planning system.
Why this matters in real flying
These are not just exam abstractions:
Radio bearings follow great circles. Radio waves travel along great circles, so a bearing from an NDB or VOR is a great-circle bearing. To plot it as a straight line on a Mercator chart, which shows rhumb lines as straight lines, you apply the conversion angle. This is a standard Radio Navigation topic.
Long-haul routes. Flight-planning systems fly great-circle segments, which is why the track displayed changes as you cross meridians even when you are flying "straight".
Chart work. On a Lambert conformal chart, a straight line approximates a great circle. On a Lambert chart, chart convergency matches earth convergency only at the parallel of origin.
Common mistakes
Mistake | Fix |
Using sin lat for departure, or cos lat for convergency | Departure is distance along a parallel and uses cos. Convergency is the angle between meridians, which grows towards the pole, so it uses sin. |
Forgetting to convert ch.long to minutes for departure | Departure in NM needs ch.long in minutes (degrees × 60). |
Using the start or end latitude instead of the mean | For two different latitudes, use the mean latitude in the convergency formula. |
Applying the full convergency instead of the CA (or the reverse) | Rhumb line ↔ great circle uses CA. GC at A ↔ GC at B uses full convergency. |
Getting the sign wrong | Sketch it. The great circle is always on the polar side. |
Errors across 000° or 180° longitude | Draw the longitudes. Between 170°E and 170°W, ch.long is 20°, not 340°. |
Treating a flight along a parallel as the shortest route | Only the equator and meridians are both rhumb lines and great circles. |
Quick-revision cheat sheet
Practice questions
These are original practice questions written for this article. They are not DGCA exam questions. Try them before checking the answers below.
Find the departure between 45°N 020°E and 45°N 030°E.
An aircraft at 60°S 100°E flies 240 NM due west. What is its new longitude?
What is the earth convergency between the meridians 060°E and 090°E at 50°N? What is the conversion angle?
Two points on the same parallel have a ch.long of 20°. The rhumb-line track is 270°(T) and the great-circle track at the start is 275°(T). Which hemisphere, and what latitude?
At what latitude does earth convergency equal ch.long, and at what latitude is it zero?
From A (30°N 010°E) to B (40°N 030°E), the initial great-circle track is 053°(T). Find the rhumb-line track and the great-circle track at B.
Answers
10° = 600′; 600 × cos 45° = 424.3 NM
Ch.long = 240 ÷ cos 60° = 480′ = 8°. Moving west from 100°E gives 092°E.
30 × sin 50° = 23.0°. CA = 11.5°
275° is north of west, so the great circle bulges north: northern hemisphere. CA = 5°, so convergency = 10°. sin lat = 10 ÷ 20 = 0.5, so the latitude is 30°N.
Equal to ch.long at the poles; zero at the equator.
Mean lat 35°: convergency = 20 × sin 35° = 11.5° and CA = 5.7°. Northern hemisphere, eastbound, so tracks increase. Rhumb line = 058.7°(T) and GC at B = 064.5°(T). Spherical trigonometry gives 058.5° and 064.7°.
FAQs
Is conversion angle always exactly half the convergency?
It is an approximation that is very accurate for normal route lengths and is the standard exam method. Over very long distances, the exact rhumb-line track drifts slightly from "initial GC track ± ½ convergency".
Which latitude should I use if the two points are at different latitudes?
Use the mean latitude in the convergency formula. For departure along a single parallel, use that parallel's latitude.
Do the formulas work if the route crosses the equator?
Cases where the route crosses the equator need extra care, because the great circle changes which pole it bends towards. Work these from a sketch rather than relying on the four-case table above.
What is the difference between earth convergency and chart convergency?
Earth convergency is the real angle between meridians on the globe. Chart convergency is the angle between meridians as drawn on a particular chart projection. They match only in specific places, depending on the projection.
Why is the great circle shorter if it looks curved on the chart?
Because most charts distort the globe. On the globe itself the great circle is the straight path. On a Mercator chart it appears curved towards the pole, while the rhumb line appears straight.
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