Learning hub · General Navigation · Lesson 6

Chart projections, great circles and convergence

The earth is round and your chart is flat, so something has to give. This lesson explains what a chart gets right and what it gets wrong, why the shortest way between two points is not the straight line you draw, and why the track you read at the start of a long leg is not the track at the end.

About 14 minutes, plus the quiz. Part of General Navigation.

1. The problem: a sphere on a sheet of paper

You cannot flatten a sphere without stretching or tearing it. Peel an orange and press the skin flat to see why. A chart maker has to choose which property to keep correct, and every choice costs something somewhere else:

• Angles and shapes (a conformal chart), so a direction measured on the chart is the true direction on the ground;
• Area, so a country keeps its size relative to others;
• Distance, which can be correct only along certain lines.

For navigation we want angles to be right, because we draw a track with a protractor and fly the heading that goes with it. Aeronautical charts for VFR flying are made conformal. The price is that the scale, the ratio of a distance on the chart to the same distance on the ground, is not the same everywhere. At any one point the scale is the same in every direction, which is what keeps shapes right, but it changes from place to place.

2. Mercator: the straight-line chart

Imagine a cylinder wrapped round the earth along the equator, with the surface projected onto it and then unrolled. The result is the Mercator chart. The meridians are parallel vertical lines, the parallels are horizontal lines, and the chart is stretched upwards in just the right amount to keep angles correct.

• The scale grows with latitude as 1 ÷ cos(latitude): it is true at the equator, 1.4 times too large at 45°, and twice too large at 60°. This is why Greenland looks huge.
• A line of constant true track, a rhumb line, is a straight line on a Mercator chart. That is what made it the sailor’s chart.
• Because of the stretching, you measure distance on the latitude scale beside the leg, not with one ruler for the whole sheet. One minute of latitude is always one nautical mile.

3. Lambert conformal conic: the aviation chart

Now wrap a cone over the earth, like a hat, cutting the surface along two parallels. These are the standard parallels. Projecting onto the cone and unrolling it gives the Lambert conformal conic chart, which is used for many ICAO aeronautical charts in the middle latitudes, including charts at 1:500,000 and 1:250,000. The legend of your chart names the projection and its standard parallels, so check it.

• Meridians are straight lines that meet at a point beyond the pole of the chart. Parallels are arcs of circles round that point.
• The scale is exactly right on the two standard parallels, a little too small between them and a little too large outside them. On a sheet with well-chosen parallels the error stays small, usually under one per cent, so one scale bar serves the whole chart.
• A straight line is very nearly a great circle, the shortest route. That is the great advantage over Mercator for flight planning.
• Because the meridians converge, a straight line crosses each one at a slightly different angle. So you measure a track at the meridian near the middle of the leg.

Try it: the same leg drawn on a Mercator chart and on a Lambert chart. Move the start and end latitude and the difference of longitude, or pick one of the legs. The orange line is the great circle and the blue line is the rhumb line. Press the Lambert box to switch charts. The grey dashes at each end are the local meridian, and the orange number is the true track on the great circle there. The numbers below the picture are worked out as you drag.

4. Reading the scale

A scale of 1:500,000 means 1 unit on the chart is 500,000 of the same unit on the ground. Chart scale is written as a ratio, so work it out in centimetres and then change to kilometres and nautical miles. One nautical mile is 1.852 km.

Scale	1 cm on the chart	1 nm on the chart
1:250,000	2.5 km, 1.35 nm	7.4 mm
1:500,000	5 km, 2.7 nm	3.7 mm
1:1,000,000	10 km, 5.4 nm	1.85 mm

A handy rule: on a 1:500,000 chart, 2 cm is about 5.4 nm, and on a 1:250,000 chart, 1 cm is a little over a mile and a third. Know the rule, then check it against the chart’s printed scale bar, because paper charts can be reprinted at a slightly different size.

5. Great circle and rhumb line

• A great circle is the line you get when a flat plane through the centre of the earth cuts its surface. It is the shortest distance between two points. The equator and every meridian are great circles.
• A rhumb line crosses every meridian at the same angle, so it is a line of constant true track. It is easy to fly, but it is longer than the great circle, except along a meridian or the equator, where the two are the same line.

On a Mercator chart the rhumb line is straight and the great circle bows towards the nearer pole. On a Lambert chart the great circle is almost straight and the rhumb line curves. Same two lines, different picture.

6. Convergence of the meridians

The meridians all meet at the poles, so they lean towards each other as you go north or south. A great circle crosses each meridian at a slightly different angle, so the true track along it keeps changing. The angle between two meridians, as it works out on the ground at a given latitude, is called the convergency:

convergency ≈ change of longitude × sin(mean latitude)

• At the equator it is zero, because the meridians are parallel there.
• At the pole it equals the change of longitude, because that is where the meridians meet.
• In between, you multiply by the sine of the mean latitude of the leg.

Over a whole great-circle leg, the true track changes by the convergency. The rhumb-line track sits half way between the track at the start and the track at the end, so the track at either end differs from it by half the convergency. That half is called the conversion angle. In the northern hemisphere the great circle lies on the pole side (north) of the rhumb line, so flying east the great-circle track grows and flying west it shrinks.

7. Why it matters so little on a VFR leg

Take a leg of about 100 nm along 45°N. It spans about 2.4° of longitude, so the convergency is about 2.4 × 0.71 ≈ 1.7°. The track at the start differs from the track at the end by under 2°, and from the rhumb line by under 1°. The distances differ by a fraction of a mile. You can hold your heading no more closely than that, and the wind correction you use is rounder than that.

The practical habits that follow:

• On a Lambert chart, draw a straight line and measure the track at the middle meridian of the leg.
• On a longer leg, expect the track to drift as you go, and check the heading against ground features rather than flying the first number blindly.
• A moving-map or GNSS unit normally computes great-circle legs, so the desired track it shows changes slowly along a long leg. This is not a fault. It is not certified for navigation here, so confirm the plan against official charts.

Convergency is a result of geometry, not of the wind or the compass. It does not change with your speed. Do not confuse it with magnetic variation, which is about where magnetic north is, not about how meridians lean.

8. A worked example

An invented leg from 50°N 010°E to 50°N 022°E, flown east.

1. The difference of longitude is 12° and the mean latitude is 50°, so the convergency is 12 × sin 50° = 12 × 0.77 ≈ 9.2°.
2. The conversion angle is half of that, 4.6°. The rhumb-line track is 090°, so the great-circle track starts at about 085° and ends at about 095°.
3. The rhumb-line distance is 12 × 60 × cos 50° ≈ 463 nm. The great-circle distance is about 462.6 nm, so the shortcut saves about half a mile. The track change is the important result, not the distance.

Set the sliders in the picture to latitude 50 at both ends and a difference of 12 to see the same numbers.

9. Try it yourself

Work each one out on paper first, then open the answer to compare. The numbers are invented.

1. A leg runs east along 60°N for 15° of longitude. What is the convergency, and what true track do you expect at the start and end of the great circle if the rhumb-line track is 090°?
   Show the answer

   Convergency ≈ 15 × sin 60° = 15 × 0.87 ≈ 13°. The conversion angle is half, about 6.5°.

   Flying east in the northern hemisphere the great-circle track grows: about 083.5° at the start and 096.5° at the end.

2. You measure 8.4 cm along a leg on a 1:250,000 chart. How long is the leg in kilometres and in nautical miles?
   Show the answer

   8.4 cm × 250,000 = 2,100,000 cm = 21 km. Divide by 1.852 to get about 11.3 nm.

   Check with the rule: 1 cm is about 1.35 nm at this scale, and 8.4 × 1.35 ≈ 11.3.

3. On a Mercator chart, why do you measure distance on the latitude scale beside the leg and not on the longitude scale along the bottom edge?
   Show the answer

   One minute of latitude is one nautical mile everywhere, and the chart stretches the latitude scale to match the stretching of the whole chart at that latitude. A minute of longitude is only a nautical mile at the equator. At 60°N it is about half a mile, because the meridians have drawn together, yet the chart still shows it as a uniform spacing. Reading distance from the longitude scale would give you the wrong answer, and a wildly wrong one at high latitude.

10. Check yourself

Five original practice questions. Answer them, press Check answers and read each explanation.

Back to the learning hub.