Convergency
Why True Track Changes as Meridians Converge
Meridians are not parallel. They meet at the geographical poles. Because the direction of True North changes from one meridian to another, the angle between a route and local True North may change as the aircraft travels east or west.
Learning objectives
By the end of this chapter, you should be able to:
- 01Define convergency
- 02Explain why meridians converge
- 03Identify the angle of convergency
- 04Calculate Earth convergency
- 05Calculate Lambert chart convergency
- 06Determine the sign of the track change
- 07Apply the DIDI mnemonic correctly
- 08Find final track from initial track
- 09Find initial track from final track
- 10Find mean track using circular averaging
- 11Solve Northern and Southern Hemisphere examples
- 12Recognise special cases and avoid common ATPL traps
Introduction and learning objectives
Convergency confuses students because it mixes three ideas: meridian geometry, great-circle geometry and chart geometry. Build them in that order and the formulae become obvious.
Meridians are not parallel. They meet at the geographical poles. Because the direction of True North changes from one meridian to another, the angle between a route and local True North may change as the aircraft travels east or west.
Convergency concerns True North and meridian geometry. It is not magnetic variation and it is not wind drift.
Meridians and True North
Move both positions and watch the two local True North arrows. They point at the same pole, but they are not parallel — and that angle is convergency.
Interactive educational diagram
Meridians, local True North and the angle between them
Move either position. Watch the two blue True North arrows: they point at the same pole but they are not parallel.
- Change of longitude
- 30.0° E
- Mean latitude
- 52.5° N
- Convergency
- 23.8°
- Track change
- +23.8°
- increase
- Initial track (great circle)
- 063° T
- Final track (great circle)
- 087° T
How it is calculated
C = Δλ × sin φm
C = 30.0 × sin 52.5° = 23.8°
The great-circle tracks shown are the exact spherical values. The formula result is the ATPL mean-latitude approximation, which is why the two can differ by a fraction of a degree on long sectors.
Definition of convergency
Three related statements of the same idea. Which one a question means depends on whether it is talking about the Earth, a great-circle track, or a chart.
Convergency is the angle between two selected meridians, measured at a specified latitude, or the change in direction of a great-circle track between two positions.
A. Meridian convergency
The angle between two meridians, evaluated at a stated latitude. On the Earth it grows with both the change of longitude and the latitude.
B. Great-circle track change
The difference between the initial true track and the final true track of one great circle. The route never bends — local True North rotates underneath it.
C. Chart convergency
The angle between two chart meridians on a projection. On a Lambert conformal conic it equals the change of longitude multiplied by the convergence factor n.
Why the true track changes
One fixed great-circle plane, one continuously rotating local meridian. Fly the route and watch the true track change while the route itself never bends.
Interactive educational diagram
Flying the great circle: the track changes, the route does not
Press play or drag the progress slider. The red great circle never bends — only the purple local meridian rotates.
- Position
- 50.0° N 20.0° W
- Current true track
- 063.0° T
- Initial true track
- 063° T
- Final true track
- 087° T
- Change so far
- 0°
- Total change
- +24.0°
The route is one fixed great-circle plane, but local True North changes along the route.
Earth convergency formula
The ATPL working formula, built step by step from the change of longitude and the signed mean latitude.
Interactive educational diagram
Interactive formula explorer — C = Δλ × sin φm
Change any coordinate and read the calculation line by line, exactly as you would write it in the examination.
Earth convergency
C = Δλ × sin φm
φm = (φ1 + φ2) / 2 · North positive, South negative
23.8°
Track change +23.8° · increase
- 1 · Positions50.0° N 10.0° W → 55.0° N 20.0° E
- 2 · Mean latitude φm(50° + 55°) / 2 = 52.5° N
- 3 · Change of longitude Δλ30.0° eastbound
- 4 · sin φm0.7934
- 5 · Convergency30.0 × 0.793 = 23.8°
- 6 · DIDI resultNorthern Hemisphere + eastbound → track increases
- Mean latitude
- 52.5° N
- Δλ
- 30.0°
- sin φm
- 0.7934
- Convergency
- 23.8°
- Hemisphere
- North
- Direction
- East
At the Equator sin 0° = 0 → C = 0°. Near the poles sin φ → 1 → C → Δλ.
Signed latitude and Equator crossing
Mean latitude uses signed values: North positive, South negative. This is where most Equator-crossing answers go wrong.
Both latitudes in the same hemisphere
40° N and 60° N → φm = +50°, i.e. 50° N. Straight average of the numbers.
Latitudes either side of the Equator
20° N and 10° S → +20 and −10 → φm = +5°, i.e. 5° N. Take the signs before averaging.
Equal opposite latitudes
30° N and 30° S → φm = 0° → convergency 0° under the mean-latitude approximation.
Both in the Southern Hemisphere
−35° and −45° → φm = −40°, i.e. 40° S. Use the magnitude in the sine and Southern DIDI for the sign.
Interactive educational diagram
Interactive formula explorer — C = Δλ × sin φm
Change any coordinate and read the calculation line by line, exactly as you would write it in the examination.
Earth convergency
C = Δλ × sin φm
φm = (φ1 + φ2) / 2 · North positive, South negative
23.8°
Track change +23.8° · increase
- 1 · Positions50.0° N 10.0° W → 55.0° N 20.0° E
- 2 · Mean latitude φm(50° + 55°) / 2 = 52.5° N
- 3 · Change of longitude Δλ30.0° eastbound
- 4 · sin φm0.7934
- 5 · Convergency30.0 × 0.793 = 23.8°
- 6 · DIDI resultNorthern Hemisphere + eastbound → track increases
- Mean latitude
- 52.5° N
- Δλ
- 30.0°
- sin φm
- 0.7934
- Convergency
- 23.8°
- Hemisphere
- North
- Direction
- East
At the Equator sin 0° = 0 → C = 0°. Near the poles sin φ → 1 → C → Δλ.
Lambert chart convergency
On a chart, the meridians converge at the rate the projection chose, not at the rate the Earth chooses. That rate is the convergence factor n.
Use the method and convergence factor specified by the question or chart data.
Interactive educational diagram
Chart convergency on a Lambert conformal conic
Move the standard parallels and the two longitudes. The meridians are drawn from the real cone constant, so their spread is the chart convergency.
- Parallel of origin (ATPL)
- 48.0° N
- n in use
- 0.7431
- n exact (secant cone)
- 0.7456
- Δλ
- 30.0°
- Chart convergency
- 22.3°
- Earth convergency at this latitude
- 22.3°
- Chart track change
- +22.3°
- Difference (chart − Earth)
- 0.0°
Chart convergency = Δλ × n, where n = sin(parallel of origin)
Do not automatically use mean route latitude when the question asks for Lambert chart convergency. Use the chart convergence factor.
Earth versus Lambert convergency
Same two positions, two different questions, two different answers. Read the wording before you pick a formula.
| Quantity | Formula | When it applies |
|---|---|---|
| Earth convergency | Δλ × sin φm | Two positions on the Earth, ATPL mean-latitude approximation. |
| Chart convergency | Δλ × n | Two meridians drawn on a specific projection. n comes from the chart, not the route. |
| Track change | TF − TI | What the aircraft actually experiences along one great circle. |
The DIDI mnemonic
Only now, with the geometry understood, is a mnemonic safe. DIDI confirms whether to add or subtract — it never replaces the reasoning.
Interactive educational diagram
The four principal cases — DIDI
Select any of the four combinations. The globe, the tracks and the sign of the change all update together.
Northern Hemisphere + eastbound → true track increases (add convergency to the initial track).
- Initial true track
- 061° T
- Final true track
- 093° T
- Convergency
- 31.1°
- Track change
- +31.1°
DIDI only tells you the sign. Always find the size of the change from C = Δλ × sin φm, then apply the sign that the hemisphere and direction demand.
Four-combination demonstrator
One tool for all four hemisphere and direction combinations, with initial, mean and final track.
Interactive educational diagram
Initial, final and mean track converter
Pick which track you were given. Try 355° with 14° of convergency to see the answer pass through 000°.
Initial track · given
355° T
Mean track
002° T
Final track
009° T
- Signed change
- +14.0°
- Half convergency
- 7.0°
- DIDI
- increase
How it is calculated
Final track = Initial track ± C
Mean track = Initial track ± C/2
Initial track = Final track ∓ C
Every result is normalised into 000°–359°, so 355° + 14° reads 009°, never 369°.
Special cases
Equator, same meridian, near the poles, the anti-meridian and equal opposite latitudes.
A. Along the Equator
Mean latitude = 0°, sin 0° = 0, therefore convergency = 0°. A route flown along the Equator has no change of true track — it is both a great circle and a rhumb line.
B. Along the same meridian
Change of longitude = 0°, therefore convergency = 0°. Tracks stay 360° or 180° throughout, whatever the latitude.
C. Near the poles
sin φ approaches 1, so convergency approaches the full change of longitude. At 90° a 25° change of longitude gives a 25° change of track.
D. Small change of longitude
Convergency is proportional to Δλ, so a 2° change of longitude at 50° N produces only about 1.5° of track change — often negligible on a short sector.
E. Crossing 180°
170° E to 170° W is a 20° change of longitude eastbound, not 340°. Convert both to signed values and take the shorter difference.
F. Crossing the Equator
Use signed latitudes in the mean: 20° N with 10° S gives φm = +5°, i.e. 5° N. Northern-Hemisphere DIDI then applies to the whole sector under the approximation.
G. Equal opposite latitudes
30° N to 30° S gives φm = 0° and therefore zero convergency under the mean-latitude approximation. The real great circle does change track either side of the Equator; the changes cancel in this symmetric case. Quote the approximation you are using.
Initial, final and mean track
Work in any direction between the three tracks and the convergency, with correct behaviour through 000°/360°.
Interactive educational diagram
Initial, final and mean track converter
Pick which track you were given. Try 355° with 14° of convergency to see the answer pass through 000°.
Initial track · given
355° T
Mean track
002° T
Final track
009° T
- Signed change
- +14.0°
- Half convergency
- 7.0°
- DIDI
- increase
How it is calculated
Final track = Initial track ± C
Mean track = Initial track ± C/2
Initial track = Final track ∓ C
Every result is normalised into 000°–359°, so 355° + 14° reads 009°, never 369°.
Track change on a Lambert chart
Drag the departure and arrival, then measure both tracks with a protractor aligned to the local meridian — not to the edge of the screen.
Interactive educational diagram
Chart convergency on a Lambert conformal conic
Move the standard parallels and the two longitudes. The meridians are drawn from the real cone constant, so their spread is the chart convergency.
- Parallel of origin (ATPL)
- 48.0° N
- n in use
- 0.7431
- n exact (secant cone)
- 0.7456
- Δλ
- 30.0°
- Chart convergency
- 22.3°
- Earth convergency at this latitude
- 22.3°
- Chart track change
- +22.3°
- Difference (chart − Earth)
- 0.0°
Chart convergency = Δλ × n, where n = sin(parallel of origin)
Do not automatically use mean route latitude when the question asks for Lambert chart convergency. Use the chart convergence factor.
Worked examples
Twelve fully explained examples, revealed one step at a time.
- GivenALDEN 50°00′N 020°00′W → BREKA 55°00′N 010°00′E, initial track 070° T
- Change of longitude30.0° eastbound
- Mean latitude52.5° N
- Convergency30.0 × sin 52.5° = 23.8°
- DIDINorth + eastbound → increase → +23.8°
- Final track070° T + 23.8° = 094° T
- Mean track070° T + 11.9° = 082° T
Practice generator
Unlimited randomly generated questions at three difficulty levels, with error tracking.
Practice mode
Randomised convergency questions
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Hemisphere and direction are stated for you. Calculate the convergency and apply DIDI visually.
Calculate the final true track for this sector.
- Departure
- 45° N 027° W
- Arrival
- 49° N 010° E
- Initial track
- 193° T
Common errors
The mistakes that cost marks in the examination, and how to avoid each one.
Using latitude instead of mean latitude
Convergency uses the mean of the two latitudes, not the departure latitude. At 40° N to 60° N use 50° N.
Forgetting to calculate the change of longitude
Δλ is a difference, not a longitude. 010° W to 020° E is 30°, not 20°.
Adding longitudes incorrectly across East and West
Same side of Greenwich → subtract. Opposite sides → add, then check the total does not exceed 180°.
Using 340° instead of 20° across the anti-meridian
Always take the shorter angular difference unless the question explicitly says the long way round.
Applying Northern-Hemisphere DIDI in the Southern Hemisphere
South of the Equator the meridians converge at the South Pole, so eastbound track DECREASES and westbound INCREASES.
Confusing initial and final track
TF = TI + signed C. If the question gives the final track, subtract the signed convergency to get back to the initial track.
Adding full convergency to get the mean track
Mean track uses HALF the convergency: TM = TI + C/2.
Adding half convergency to get the final track
Final track uses the FULL convergency.
Using Earth convergency when the question asks for chart convergency
A Lambert question wants Δλ × n from the chart data. Mean route latitude is irrelevant to the chart.
Using mean route latitude instead of the Lambert convergence factor
n is fixed by the standard parallels for the whole chart; it does not change with your route.
Forgetting to normalise tracks
350° + 20° is 010°, not 370°. Always bring the answer back into 000°–359°.
Treating meridians as parallel
Parallel meridians would mean zero convergency everywhere — which is only true on a Mercator chart, and only for the chart, never for the Earth.
Confusing magnetic variation with convergency
Variation is the angle between True and Magnetic North at one place. Convergency is between True North at two places.
Confusing track change with wind correction angle
Convergency changes the TRACK. Drift changes the relationship between heading and track. They are independent.
Knowledge check
Question 1 of 22 · Multiple choice
Meridian geometry
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Chapter summary
- 1Meridians meet at the polesThey are not parallel anywhere except at the Equator crossing point of view.
- 2Local True North changesEach meridian defines its own True North direction.
- 3Great-circle true track changesThe plane is fixed; the reference direction rotates.
- 4Earth convergencyC = Δλ × sin mean latitude
- 5Lambert chart convergencyC = Δλ × convergence factor n
- 6DIDINorth westbound = Decrease · North eastbound = Increase · South westbound = Increase · South eastbound = Decrease
- 7Final trackInitial track ± convergency
- 8Mean trackInitial track ± half convergency
Understand the meridians first. Use DIDI only to confirm the sign.
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