Convergency and Chart Projections

You plan a flight from Shannon (50°N 010°W) to Moscow (50°N 040°E) and draw the same straight pencil line on two charts side by side. On the Mercator the line looks reassuringly direct — east, parallel to the parallel of latitude. On the Lambert the line tilts up over the top of the page, bowing northward through Iceland and Sweden. Same flight, two charts, two completely different shapes. Which one shows the shortest route?

The answer hinges on a single bit of spherical geometry that every ATPL exam keeps coming back to: convergency, the angle by which meridians lean towards each other as they run to the nearer pole.

Formal definitions

Each of these is taken straight from Oxford ATPL General Navigation, chapter 14 (Convergency and Conversion Angle) and chapter 18 (Mercator Charts — Properties). Read the quote first; the plain-English line below is just a paraphrase to make sure the definition stuck.

Earth convergency

Convergency is defined as the angle of inclination between two selected meridians measured at a given latitude. — Oxford ATPL General Navigation, Ch. 14, p. 241

In plain terms: two meridians look parallel at the equator, meet at the pole, and lean towards each other by some intermediate angle in between. That lean is the convergency.

Chart convergency

The angle of inclination between meridians on the chart, (or the change in direction of a straight line), between 2 longitudes. — Oxford ATPL General Navigation, Ch. 18, p. 299

Earth convergency is a property of the Earth. Chart convergency is a property of the chart. The two only agree if the chart projects the meridians honestly. On a Mercator the meridians are drawn parallel, so chart convergency is zero everywhere — even at the pole, where the Earth's meridians actually meet head-on.

Rhumb line

A Rhumb Line is a regularly curved line on the surface of the Earth which cuts all meridians at the same angle — a line of constant direction. — Oxford ATPL General Navigation, Ch. 2, p. 24

The line your compass would draw if you held one heading and never turned. On a globe it spirals towards the pole; on a Mercator it is a straight line (which is the entire reason Mercator built the projection in the first place).

Great circle

A great circle is any circle on the Earth whose centre and radius are those of the Earth itself. The shortest path between two points on the sphere is always an arc of a great circle. Because each meridian intersection rotates the local "north" direction, the great-circle track angle changes continuously along the route.

Conversion angle

Conversion angle is the difference between great circle direction and Rhumb Line direction joining two given points. Conversion angle is ½ convergency. — Oxford ATPL General Navigation, Ch. 14, p. 251

The track you'd read off a chart for the same two endpoints differs between the two routes by exactly half the convergency, and that difference is the same at both ends.

The three projection families

Mercator: meridians parallel

A Mercator projection wraps a cylinder around the equator and projects the Earth outwards onto it. After unrolling, the meridians come out as equally-spaced parallel lines. So:

• A straight line on a Mercator is a rhumb line — it cuts every vertical meridian at the same angle.
• The great circle between the same two points must therefore look curved, bowing towards the nearer pole.
• Chart convergency is zero; this matches Earth convergency only at the equator.

The bow is real geometry, not a quirk of the printer. The great-circle arc is the shorter route — it just looks longer on a Mercator because the projection stretches the higher latitudes outward.

Scale on a Mercator is not constant. It is correct only at the equator and expands as sec(latitude). A chart advertised as 1:1 000 000 at the equator is really 1:500 000 at 60° latitude — twice as large. Above ~70° the expansion becomes so extreme that the chart is unusable, which is why Mercator is the equatorial-belt chart and Polar Stereographic takes over near the poles.

Lambert: meridians converge to the parallel of origin

A Lambert conformal conic is built by sitting a cone over the Earth so that it touches at one parallel of origin (or, in the modified two-standard-parallel form, cuts at two standard parallels with the parallel of origin halfway between). Cut the cone open and lay it flat; the meridians come out as straight lines radiating from the apex of the cone.

chart convergence = change of longitude × sine of parallel of origin — Oxford ATPL General Navigation, Ch. 21, p. 342

So on a Lambert:

• A straight line drawn on the chart is almost a great circle — exact at the parallel of origin, very close elsewhere.
• The rhumb line between the same two points is the curved one, and it bends towards the equator.
• Chart convergency matches Earth convergency exactly at the parallel of origin and is a very close approximation across the rest of a typical chart.

This is why every ICAO 1:500 000 topographical chart you fly with is a Lambert conformal conic — straight-line plotting on a Lambert is honest great-circle navigation to within a fraction of a degree.

The formulas

For both definitions, mean latitude means the arithmetic mid-latitude between the two points (Oxford notes that the true mean is slightly nearer the higher-latitude pole, but the error is negligible for the ranges normally examined).

• Earth convergency = change of longitude × sin(mean latitude)
• Conversion angle = ½ × change of longitude × sin(mean latitude)
• Chart convergence (Lambert) = change of longitude × sin(parallel of origin)
• Chart convergence (Mercator) = 0

At the equator, sin(0) = 0, so convergency vanishes — meridians on the Earth genuinely are parallel there. At the pole, sin(90°) = 1, so convergency equals the full change of longitude — meridians cross at exactly that angle.

Worked example

Flight from A (50°N 010°W) to B (50°N 040°W) along the same parallel of latitude.

Step 1 — Change of longitude. Both points are west of Greenwich, so: ch.long = 040°W − 010°W = 30°

Step 2 — Mean latitude. Both points are at 50°N, so the mid-latitude is 50°N. sin(50°) = 0.766

Step 3 — Earth convergency. convergency = 30° × sin(50°) = 30 × 0.766 = 22.98°

Step 4 — Conversion angle. CA = ½ × convergency = ½ × 22.98° = 11.49°

Step 5 — Translate into track angles. The rhumb-line track is exactly 270°(T) the whole way (you're flying due west along the parallel). The great circle bows toward the nearer pole (north, in this hemisphere), so it leaves A heading slightly north of due west: 270° + 11.49° = 281.49°(T) …and, having crossed the bow, arrives at B heading slightly south of due west: 270° − 11.49° = 258.51°(T)

The change of great-circle track between A and B is 281.49° − 258.51° = 22.98° — exactly the convergency, as the definition requires.

Common mistakes

1. Confusing convergency with conversion angle. Convergency is the full angle between the two meridians (and the full change in great-circle track). Conversion angle is half that. If you forget the factor of two, every great-circle bearing you compute will be out by 50%.
2. Believing the Mercator great-circle "wave" is a chart artefact only. The wave is drawn by the projection, but the path it represents is the genuine shorter route over the Earth. Pilots who refuse to fly the curve because "the straight line on the Mercator looks more direct" burn extra fuel for nothing.
3. Forgetting that convergency = 0 at the equator. Anywhere along the equator, mean latitude is 0°, sin(0) = 0, and rhumb-line and great-circle tracks coincide. There is no conversion angle, and straight-line plotting on a Mercator is exact.
4. Using "mean latitude" for trans-equatorial flights. If A is 30°N and B is 30°S, the simple arithmetic mean is 0°, which would give a convergency of zero — but the great circle definitely is not a parallel of latitude. For trans-equatorial flights, split the leg at the equator and treat each hemisphere separately.

Why it matters

Almost every chart-work question on the ATPL General Navigation paper hides a convergency calculation. "Initial great-circle track from A to B is 060°(T) — what is the great-circle track from B to A?" is just asking you to add the convergency and take the reciprocal. "Convert the rhumb-line track on this Mercator into the initial great-circle bearing the FMC will fly" is conversion angle, nothing more. The cockpit choice of chart — Mercator for the equatorial belt, Lambert for the mid-latitudes, Polar Stereographic above 70° — is a direct consequence of how each projection handles convergency.