Great circles and the shortest route
A great circle is the Earth geometry behind the shortest surface route. Its plane passes through the Earth's centre, so its centre and radius are the same as the Earth's.
A reminder about great circles
Imagine cutting the Earth with a flat plane. If that plane passes through the centre of the Earth, its intersection with the surface is a great circle. A disc cut in this plane has the largest possible area of any circular section through the Earth. If the plane misses the Earth's centre, it produces a small circle.
One route through two points
Two surface points and the Earth's centre normally define one plane, so only one great circle passes through the two points. The circle offers two arcs between them. The shorter arc is the shortest distance over the spherical surface.
The exception is a pair of antipodal points. They lie at opposite ends of a diameter. Infinitely many planes can pass through that diameter, so infinitely many great circles join the pair. Every half-circle route between antipodes is 180 degrees or 10 800 NM on the spherical navigation model.
Familiar great circles
The Equator is a complete great circle. A meridian is a semi-great circle running from pole to pole. A meridian and its anti-meridian together form one complete great circle. These special cases matter because their angular graduations can be converted directly into distance.
Vertices
A non-equatorial great circle has two vertices, opposite each other. A vertex is the point of greatest latitude reached by that great circle, one in each hemisphere. At a vertex the great circle is tangent to a parallel of latitude, so its instantaneous direction is 090 degrees or 270 degrees. A polar great circle has its vertices at the poles.
Small circles
A small circle is formed when the cutting plane does not pass through the Earth's centre. Every parallel of latitude except the Equator is a small circle. Its radius is smaller than the Earth's radius and becomes smaller towards either pole.
Lines which are both great circles and rhumb lines
The only coincident cases are the Equator and a meridian route. The Equator is a complete great circle and a 090 or 270 degree rhumb line. A single meridian is a semi-great circle and a 000 or 180 degree rhumb line; paired with its anti-meridian, it completes a great circle.
| Surface line | Circle type | Navigation property |
|---|---|---|
| Equator | Great circle | Also a rhumb line |
| Single meridian, northbound or southbound | Semi-great circle | Also a rhumb line, cut angle 0 degrees |
| Meridian plus anti-meridian | Complete great circle | Passes through both poles |
| Parallel other than the Equator | Small circle | A rhumb line, cut angle 90 degrees |
| General great circle | Great circle | Shortest route, direction normally changes |
The rhumb line
A rhumb line trades minimum distance for a constant direction. It crosses every meridian at the same angle and is therefore also called a loxodrome.
The rhumb line
A rhumb line is a regularly curved line on the Earth that cuts every meridian at the same angle. A pilot following it can maintain one track direction, apart from corrections for wind and changes in magnetic reference. Before computer navigation, this constant-direction property made a rhumb line much easier to follow than a changing great-circle track.
For the normal route without extra revolutions around the Earth, one rhumb line joins two specified non-polar points. This is the constant-direction alternative to the one great circle through the same points.
Its shape on the globe
Except for the Equator and meridians, a rhumb line curves on the globe. Continued indefinitely towards a pole, it spirals around the pole and approaches it without arriving. On a normal Mercator chart the increasing spacing of the parallels straightens this spiral, so every rhumb line plots as a straight line.
Examples of rhumb lines
- Every parallel of latitude cuts the meridians at 90 degrees.
- The Equator is a parallel and is also a great circle.
- Every meridian cuts the other meridians only at the poles and represents a constant north or south direction. A meridian with its anti-meridian is also a great circle.
Rhumb line compared with great circle
For two ordinary points, the rhumb line is longer than the shorter great-circle arc. The two distances are equal only when the common route lies along the Equator or along a meridian. The rhumb line between two points lies on the equatorial side of the corresponding great circle. In the Northern Hemisphere it lies south of the great circle. In the Southern Hemisphere it lies north of it.
| Feature | Great circle | Rhumb line |
|---|---|---|
| Distance | Shortest surface distance | Normally longer |
| Direction | Normally changes along the route | Constant |
| Relative position | On the polar side | On the equatorial side |
| At a pole | May pass through it | Spirals towards it unless following a meridian |
| Equal routes | Only along the Equator or a meridian | |
Great-circle direction and conversion angle
A great circle is shortest, but its angle with successive meridians normally changes. That changing angle is the practical difference between a great-circle track and a rhumb-line track.
Great-circle direction
Track direction is measured clockwise from the local meridian. Since meridians converge towards a pole, a great circle that is not the Equator or a meridian cuts successive meridians at different angles. A modern navigation system can continuously update the commanded track and keep the aircraft close to the great circle.
On an eastbound great circle in the Northern Hemisphere, the true track normally increases until the vertex and continues to increase after it. In the Southern Hemisphere an eastbound track normally decreases. Reverse the direction of travel and the numerical change reverses. A quick memory pattern for the four cases in order, Northern westbound, Northern eastbound, Southern westbound, Southern eastbound, is decreasing, increasing, increasing, decreasing.
| Hemisphere | Direction of travel | Change in great-circle track |
|---|---|---|
| Northern | Westbound | Decreasing |
| Northern | Eastbound | Increasing |
| Southern | Westbound | Increasing |
| Southern | Eastbound | Decreasing |
The vertex and the midpoint track
At a vertex the great circle is tangent to a parallel. The local great-circle direction is therefore east or west. On a symmetrical route between equal latitudes, the vertex lies halfway in longitude and the track there matches the mean rhumb-line direction.
A first look at conversion angle
The conversion angle is the angular difference used when changing between a great-circle direction and the corresponding rhumb-line direction at a point. It accounts for the change of great-circle direction caused by meridian convergence. For the usual short treatment, conversion angle is half the convergency between the end meridians.
Suppose the convergency between two meridians is 12 degrees. The conversion angle is 6 degrees. The initial and final great-circle tracks lie approximately 6 degrees on opposite sides of the constant rhumb-line track for a symmetrical case.
The shorter great-circle arc is the shortest distance between two surface points.Oxford ATPL Book 10, chapter 2
Great-circle distance is measured as change in latitude along a meridian, while east-west distance on a parallel is departure.R.K. Bali, Air Navigation, chapter 7
Distance units used in navigation
The nautical mile links a linear distance to the angular graticule. The fixed ICAO nautical mile is exact, while a geographic minute of arc on the real oblate Earth varies slightly with latitude.
Distance on the Earth
Aviation uses both metric and imperial measures, but route distance is normally expressed in nautical miles because that unit connects directly to angular measure on the Earth's graticule.
The nautical mile
The historical geographic idea is the length of a great-circle arc that subtends one minute at the centre of curvature. The operational international unit is fixed: one nautical mile is exactly 1852 metres. Traditional navigation texts also use 6080 ft as the standard or Admiralty nautical mile. The two values are close but not exactly identical.
Variations in the length of a nautical mile
The Earth is an oblate spheroid and latitude is geodetic. Its radius of curvature changes from equator to pole, so the surface arc represented by one minute of latitude is not perfectly constant. Bali's table prints rounded illustrative values of about 6040 ft at the Equator, 6080 ft at 45 degrees and 6170 ft near the poles. Oxford gives about 6048 ft at the Equator and 6108 ft at the poles. For operational conversion, the fixed ICAO value of 1852 m is controlling.
The kilometre
The metre is the SI base unit of length. Historically, the kilometre was related to one ten-thousandth of the mean meridional distance from Equator to pole. The rounded Earth model therefore gives 10 000 km from Equator to pole and 40 000 km around the Earth.
The statute mile and smaller units
A statute mile is 5280 ft. It remains common for ground distance and in a few national conventions, but aviation navigation normally uses nautical miles. One foot is 12 inches, one yard is 3 ft and one inch is exactly 2.54 cm.
Conversion factors
| From | To | Conversion |
|---|---|---|
| 1 NM | kilometres | 1.852 km |
| 1 km | nautical miles | 0.540 NM approximately |
| 1 NM | statute miles | 1.151 statute miles approximately |
| 1 statute mile | nautical miles | 0.869 NM approximately |
| 1 m | feet | 3.281 ft approximately |
| 1 ft | metres | 0.3048 m |
| 1 inch | centimetres | 2.54 cm exactly |
| 1 m | centimetres and millimetres | 100 cm or 1000 mm |
| 1 cm | millimetres | 10 mm |
| 1 yard | feet | 3 ft |
| 5400 NM | kilometres | 10 000 km on the rounded Earth model |
| 21 600 NM | kilometres | 40 000 km on the rounded Earth model |
Distance along a meridian or the Equator
The graticule becomes a distance scale on special great circles. Minutes of latitude work along every meridian. Minutes of longitude work directly only on the Equator.
Great-circle distances
General great-circle distance requires spherical trigonometry, but the examination cases here place both points on a meridian, a meridian and anti-meridian pair, or the Equator. These are solved directly from angular distance.
Distance on the same meridian
Along a meridian, one minute of change of latitude is taken as one nautical mile. Convert the angular separation into minutes, then read the same number as nautical miles.
Same hemisphere example
From 51 degrees 37 minutes north to 6 degrees 48 minutes north on the same meridian, change of latitude is 44 degrees 49 minutes. Distance is 44 multiplied by 60 plus 49, giving 2689 NM.
Opposite hemispheres example
From 29 degrees 30 minutes south to 59 degrees 47 minutes north on the same meridian, add the latitudes. The change is 89 degrees 17 minutes, so the distance is 5357 NM.
Distance along the Equator
The Equator is a great circle. One minute of longitude measured along it is one nautical mile. From 16 degrees 35 minutes west to 103 degrees 55 minutes east, the shorter change of longitude is 120 degrees 30 minutes. The distance is 120 multiplied by 60 plus 30, giving 7230 NM.
Choosing the smaller longitude arc
When the direct difference in longitude exceeds 180 degrees, subtract it from 360 degrees to obtain the smaller angular separation and reverse the east or west sense. For example, a direct difference of 303 degrees 18 minutes corresponds to the shorter change of longitude of 56 degrees 42 minutes.
| Where the points lie | Angular separation | Distance rule |
|---|---|---|
| Same meridian, same hemisphere | Difference of latitudes | Minutes of latitude equal NM |
| Same meridian, opposite hemispheres | Sum of latitudes | Minutes of latitude equal NM |
| Both on the Equator | Smaller change of longitude | Minutes of longitude equal NM |
| Arbitrary parallel | Change of longitude | Must allow for the parallel's smaller radius |
Polar routes, antipodes and mean latitude
Meridian and anti-meridian pairs form one great circle. Their shortest arc may pass over a pole, and that possibility must be drawn before calculating the distance.
Meridian and anti-meridian
Two longitudes are a meridian and anti-meridian when their numerical values add to 180 degrees if their names are opposite. For 11 degrees 10 minutes east and 168 degrees 50 minutes west, the sum is 180 degrees.
Same hemisphere
For 41 degrees 55 minutes north and 21 degrees 17 minutes north on opposite halves of the same meridional great circle, the shorter route goes over the North Pole. Its angular length is 180 degrees minus the sum of the latitudes. This gives 116 degrees 48 minutes, or 7008 NM.
Different hemispheres
For 35 degrees 57 minutes north and 22 degrees 10 minutes south on a meridian and anti-meridian, compare the two polar routes. Via the North Pole the arc is 166 degrees 13 minutes, or 9973 NM. A proposed arc of 193 degrees 47 minutes is the longer way round. Subtract it from 360 degrees to recover the shorter 166 degrees 13 minutes.
Antipodal points
Equal latitudes in opposite hemispheres and longitudes differing by 180 degrees describe antipodal points. Their angular separation is exactly 180 degrees. Their spherical distance is 180 multiplied by 60, which is 10 800 NM, half the 21 600 NM circumference.
Mean latitude
Mean latitude is used in several short-range navigation approximations. When both latitudes have the same name, add them and divide by two. Between 52 degrees 17 minutes north and 17 degrees 57 minutes north, the sum is 70 degrees 14 minutes and the mean is 35 degrees 07 minutes north.
When the latitudes have contrary names, subtract the smaller from the larger and divide the difference by two. The mean lies in the hemisphere of the larger latitude. Between 35 degrees 25 minutes north and 13 degrees 38 minutes south, the algebraic mean is 10 degrees 53.5 minutes north.
| Case | Shortest angular arc | Example result |
|---|---|---|
| Meridian and anti-meridian, same hemisphere | 180 degrees minus the two latitudes | 116 degrees 48 minutes, 7008 NM |
| Meridian and anti-meridian, different hemispheres | Compare routes through both poles | 166 degrees 13 minutes, 9973 NM |
| Antipodes | 180 degrees | 10 800 NM |
Distance along a parallel and final comparison
A parallel is a smaller circle away from the Equator. Its east-west distance is called departure, so longitude minutes must be reduced by the cosine of latitude.
Rhumb-line distance along a parallel
The radius of a parallel at latitude L is the equatorial radius multiplied by cosine L. The same fraction applies to an arc measured along that parallel. Bali therefore writes the east-west rhumb-line distance as departure.
At 27 degrees 21 minutes north, a change of longitude of 78 degrees 23 minutes is 4703 minutes. Multiplying by cosine 27 degrees 21 minutes gives approximately 4177 NM. Chapter 5 develops departure and its inverse in full.
Why the special cases agree
At the Equator, cosine latitude is 1, so departure equals change of longitude in minutes. At either pole, cosine latitude is zero, so every longitude meets and departure for any longitude change is zero. This is why the one-minute rule for longitude is exact only at the Equator.
Great circle or rhumb line
- Choose a great circle when the minimum surface distance matters.
- Expect the true track to change unless the route follows the Equator or a meridian.
- Choose a rhumb line when one constant direction is operationally useful.
- Remember that the rhumb line stays on the equatorial side and is normally longer.
- Use conversion angle to relate the two bearings. Chapter 6 completes that method.
| Exam cue | Immediate response |
|---|---|
| Shortest surface route | Shorter arc of the great circle |
| Constant direction | Rhumb line |
| Every meridian at the same angle | Rhumb line definition |
| Parallel other than Equator | Small circle and rhumb line |
| Minute of longitude equals 1 NM | Only on the Equator |
| Opposite points | Antipodes, 180 degrees, 10 800 NM |