Great Circle Map: Find Your Beam Heading and Distance for DX

The first time you point a beam at a station in Japan, you may be surprised. On a flat wall map Japan lies east of Europe, so you aim east, yet the signal is strongest when the antenna points northeast. The explanation is that radio waves follow the shortest path over the surface of the Earth, and on a flat map that shortest path is a curve. A great circle map corrects the picture. It shows the true direction and the true distance from your station to anywhere in the world, so you can set a rotator, decide between short and long path and understand why certain openings happen when they do. This guide explains the idea, the maths behind the bearing and the distance, and how to use the result at the operating position.

What a Great Circle Is

On a sphere, a great circle is a circle whose centre is the centre of the sphere. The equator and every line of longitude are examples, and so is any circle you get by slicing the sphere through its middle. As the great circle article explains, the shortest path between two points on the surface of a sphere runs along a great circle. The Earth is not a perfect sphere, but for beam headings and DX distances the sphere is accurate enough, and the errors are far smaller than the width of an antenna beam.

Two points that are not opposite each other define a unique great circle, and that circle is divided into two arcs. The shorter arc is the short path. The longer arc is the long path, which goes the other way around the planet. The two bearings from your station are exactly 180 degrees apart, and the two distances add up to the full circumference of the Earth, which is about 40000 kilometres.

Why a Flat Map Misleads

Every flat map distorts something. The familiar Mercator projection, used in most wall maps and online map viewers, keeps angles correct locally, which is excellent for sailing along a constant compass course, but it stretches areas and distances away from the equator, and it turns great circles into curves. A straight line drawn on a Mercator map is not the shortest route, so a ruler and a protractor give the wrong heading for a long path. The error is small for a contact over a few hundred kilometres. For a DX path across a continent or an ocean it can be tens of degrees, enough to miss the other station entirely with a narrow beam.

That is why a station in northern Europe aims northeast to reach Japan and northwest to reach the east coast of North America, even though both look due east or due west on a flat map. The great circle arcs towards the pole because meridians converge there, and the shortest route bends accordingly.

The Azimuthal Equidistant Map

The best map for a radio amateur is the azimuthal equidistant projection centred on your own location. As the azimuthal equidistant projection article states, all points on such a map are at proportionally correct distances from the centre, and at the correct azimuth, or direction, from the centre point. A straight line from the centre to any place is therefore a great circle, and the direction of that line is the beam heading. The distance along the line is the great circle distance.

The limit of the projection is that the distances and directions are correct only from the centre. Between two other points on the map the picture is distorted, so it is a map for your station and not for the whole world. If you move or want a map for another location, you need a new map centred there. The edge of the map, at 20000 kilometres, is a single point on the opposite side of the globe, your antipode, and the whole outer ring of the picture represents that one point. Land masses on the far side of the world look very stretched.

Several kinds of map can help with different jobs.

Calculating Bearing and Distance

You do not have to rely on a printed map. If you know the latitude and longitude of both stations, two short formulas give the answer. Let the latitude and longitude of your station be phi1 and lambda1, those of the other station phi2 and lambda2, and let delta lambda be the difference in longitude. The initial bearing, measured clockwise from true north, is the angle whose tangent is the ratio of two terms: sin(delta lambda) times cos(phi2) in the numerator, and cos(phi1) times sin(phi2) minus sin(phi1) times cos(phi2) times cos(delta lambda) in the denominator. Use the two-argument arctangent so the answer lands in the right quadrant, then add 360 degrees if the result is negative.

The distance uses the haversine formula. Compute a as sin squared of half the latitude difference plus cos(phi1) times cos(phi2) times sin squared of half the longitude difference. The central angle is twice the arcsine of the square root of a, and the distance is that angle in radians multiplied by the radius of the Earth, about 6371 kilometres. Spreadsheets and logging programs do this for you, but it is useful to know what they are doing, and to be able to check one when a result looks wrong.

The positions need not be exact. Your locator square is enough for a rough heading. The Maidenhead system is described in our guide to grid squares and the Maidenhead locator, and a six-character locator gives a position within a few kilometres, which is more than good enough for a beam.

Worked Example from Vilnius

Take a station near Vilnius, at roughly 54.7 degrees north and 25.3 degrees east. The following figures are approximate, calculated with the formulas above for the centre of each target city.

• Tokyo. Short path bearing about 50 degrees, distance about 8200 kilometres. The long path bearing is about 230 degrees, at roughly 31800 kilometres.
• Wellington, New Zealand. Short path bearing about 69 degrees, distance about 17300 kilometres. The long path bearing is about 249 degrees, at roughly 22700 kilometres. Both paths are very long, which is typical of near-antipodal paths.
• Washington, DC. Short path bearing about 303 degrees, distance about 7300 kilometres.
• Cape Town. Short path bearing about 186 degrees, distance about 9900 kilometres.

Notice that Tokyo lies at 50 degrees, northeast, while a Mercator map would suggest something much closer to east. Washington is 303 degrees, northwest, not west. These differences are the reason that a naive flat map, or a guess, leaves the beam pointing in the wrong direction.

Long Path and Short Path

Because the two bearings are 180 degrees apart, turning the antenna around is the same as switching to the long path. The signal on that path travels much farther, and it passes through different parts of the ionosphere, often along the grey line where absorption is lower. Openings on the long path can be strong when the short path is closed, especially near sunrise and sunset. Our article on long path DX explains when it happens and how to recognise it. A great circle map makes it easy to see which stations might be reached by turning the beam, and which part of the world lies in the beam on each path.

Remember that on a near-antipodal path, a small difference in position changes the bearing a lot. Directions near the antipode are very sensitive, so expect signals to arrive from a wide range of angles, and be ready to try several headings.

True North, Magnetic North and the Rotator

The bearings calculated above are relative to true north. A magnetic compass shows magnetic north, and the difference, called magnetic declination, varies with the place and changes slowly with time. If you set the beam with a handheld compass, correct for the declination. Most rotator controllers are calibrated to a compass dial, and the most reliable approach is to check the calibration against a known reference, such as the position of the sun at a computed time, a distant landmark, or a station whose bearing you know. Our guide to the antenna rotator covers calibration, stops and computer control, and shows how logging programs send the computed bearing directly to the rotator.

Beam width matters too. A three element Yagi has a broad pattern, and a heading that is 10 or 15 degrees off makes little difference. A large array has a narrow beam, and the same error can cost many decibels. When you pick between a Yagi and other directional antennas, see our comparison of a Yagi and a hex beam, and remember that the narrower the beam, the more accurate your map must be.

Using Great Circle Maps in Practice

• Print or display a centred map. Many websites and programs can draw an azimuthal map centred on your locator, with a ring for each 2000 or 5000 kilometres and a line for each 10 or 30 degrees.
• Mark your favourite targets. Draw the bearings of the regions you most want to work, such as your target countries in a contest or the areas where you expect an opening.
• Use the logger. Most logging programs show the bearing and the distance as soon as you type a callsign, using the country or the locator. Rely on that, but understand what it displays.
• Check both paths. When the short path sounds dead, try the long path before you give up.
• Use it for planning. If you know the bearing to a rare country, you can decide which antenna to use, and at what hours the path is likely to be open.
• Combine it with a day and night map. If the path lies mostly in darkness, low-band propagation is likely. If it crosses the terminator, you may find enhanced propagation.

Common Mistakes

The most frequent mistake is to use a flat map and read a heading with a protractor. The second is to use a map centred on the wrong place, such as a world map centred on a capital city that is not near you. The third is to forget that the bearing from A to B is not the reverse of the bearing from B to A, except by 180 degrees on a very short path. On a long great circle the direction changes along the way, so the other station’s heading to you is different from yours to them. The fourth is to confuse magnetic and true north. And the fifth is to trust the beam heading when the rotator is poorly calibrated. A calibrated rotator and a correct map together make DX easier.

The Bottom Line

A great circle map is the honest picture of where your signals go. Centred on your own station, it shows the true direction and distance to every place on Earth, and it reveals why a path to Japan runs northeast and why a path to the Pacific can be shorter the other way around. Learn the formulas so you can trust your software, set up a rotator you can calibrate, remember the long path and keep true north separate from magnetic north. Once you start thinking in great circles, the maps in your shack and the headings of your beam finally agree.

Frequently Asked Questions

What is a great circle?

It is a circle on a sphere whose centre is the centre of the sphere. The shortest path between two points on the surface follows a great circle.

Why is the beam heading to Japan northeast from Europe?

The shortest route follows a great circle that bends towards the north pole. A flat map does not show this, so the straight line east on the map is not the shortest path.

What is the best map for beam headings?

An azimuthal equidistant map centred on your own location. Straight lines from the centre are great circles, and their direction and length are true.

How do I find the long path heading?

Add or subtract 180 degrees from the short path bearing. The long path distance is about 40000 kilometres minus the short path distance.

Do I need to correct for magnetic declination?

Yes, if you use a magnetic compass. Calculated bearings are relative to true north, and the declination at your location is the difference between true and magnetic north.

How accurate does the heading have to be?

It depends on the beamwidth. A small Yagi tolerates errors of 10 degrees or more, while a large array needs a heading accurate to a few degrees.