Great Circles and Rhumb Lines - ATPL 061 — General Navigation

The first lesson of this course noted in passing that a meridian and its anti-meridian together complete a full circle around the Earth. This lesson is where that fact grows into a subject: the geometry of circles on a sphere, and what it costs, in miles and in degrees, to fly one kind of line rather than the other.

Great circles and small circles

Cut a sphere with a plane and the cut is always a circle; which kind depends on where the plane passes:

a great circle is the intersection of the Earth's surface with a plane passing through the Earth's centre: it is the largest circle that can be drawn on the sphere, it divides the Earth into two equal halves, and through any two points that are not diametrically opposite each other runs exactly one great circle;
a small circle is the cut of any plane that does not pass through the centre: every parallel of latitude except the equator is one.

A great circle that is not the equator and not a meridian reaches its vertex at the highest latitude it attains, one in each hemisphere, and at the vertex its direction is exactly east-west, 090 or 270 true. The two vertices lie 180° of longitude apart, and between them the circle crosses the equator twice. The Earth's ready-made examples of great circles are the equator and every meridian taken together with its anti-meridian; the parallels, cut by planes north or south of the centre, are the standing examples of small circles.

The great circle's claim to fame is economy: the shorter arc of the great circle through two positions is the shortest distance between them on the Earth's surface. The reason is the geometry of the sphere itself: of all the circles through the two points, the great circle is the one with the largest radius, and the flattest curve between two points is the shortest path a curved surface allows. Stretch a string tight between two points on a globe and it settles onto the great circle by itself.

The rhumb line

The alternative is built for the compass rather than the fuel gauge. A rhumb line (loxodrome) is a line on the Earth's surface cutting all meridians at the same angle: a line of constant true direction. Held long enough away from the special cases, it spirals ever tighter toward the pole without ever quite reaching it. The meridians, the equator and the parallels of latitude are all rhumb lines, and of them only the equator and the meridians run along great circles, a meridian pairing with its anti-meridian to complete one. Everywhere else, a rhumb-line route is not the shortest distance between two positions: its price for the constant heading is extra miles.

Convergence

Why can the great circle not simply hold one direction too? Because the reference it is measured against will not stand still. Direction is measured from the meridians, the meridians converge toward the poles, and a great-circle track therefore cuts each successive meridian at a different angle: its true direction changes continuously along the route, the equator and the meridians themselves being the only exceptions. The total change between two positions is the Earth convergence, and the syllabus approximates it with one formula:

One worked instance inside the up-to-30°-of-longitude range the syllabus calculates in: between meridians 20° apart at a mean latitude of 50°N, the convergence is 20 × sin 50°, which is 20 × 0.766, about 15°. Stretched to the figure's wider crossing, 40° of longitude at mean latitude 45°N, the same formula gives 40 × 0.707, about 28°. At the equator the sine is zero and the meridians run parallel: no convergence, which is exactly why the equator can be both straight-in-direction and shortest at once.

The conversion angle

Between the same two points the great circle and the rhumb line leave in different directions, and the angle between them has its own name and a very tidy size: the conversion angle is the angle between the great-circle bearing and the rhumb-line bearing, and it equals half the convergence:

Applying it needs one picture held firmly: the great circle always lies on the poleward side of the rhumb line, so in the northern hemisphere the great-circle track out of a position points closer to the pole than the rhumb-line track by the conversion angle. On the 45°N crossing above (convergence 28°, conversion angle 14°): the rhumb line runs a constant 090; the great circle leaves at 090 minus 14, which is 076, passes its vertex doing exactly 090, and arrives at 090 plus 14, which is 104. The rhumb line splits the difference the whole way, which is precisely why the conversion angle is half the convergence.

How much the bow is worth depends on where you fly it: the difference in distance between the great-circle and rhumb-line routes increases with latitude and with the change of longitude. Near the equator the two lines almost coincide; on short legs the difference is pennies. It is the long, high-latitude, east-west crossing where the great circle earns its keep, and the example below prices one.

What the exam asks

The two formulas and the poleward-side rule carry the calculation marks; the definitions of vertex, great circle and rhumb line carry the rest. The overview, following the learning objectives:

Describe the geometric properties of great and small circles including the vertex; explain why the great circle is shortest; name examples on the Earth

Describe the rhumb line's geometry and state that it is not the shortest route

Explain why a great-circle track changes direction; state and apply convergence ≈ Δlong × sin(mean latitude) up to 30° of longitude

Calculate and apply the conversion angle, and explain how the great-circle vs rhumb-line distance difference varies with latitude and change of longitude

Great circles and small circles

The rhumb line

Convergence

Check you understood

The conversion angle

What the exam asks