On an ocean the straight line on your chart is not the shortest way. Find the great circle, sail it as waypoints, keep it out of the ice, and know when a longer route is faster.
1 · Great circle and rhumb line
Shortest track or steady course
Why is the straight line on a Mercator chart not the shortest route across an ocean?
Because that straight line is a rhumb line: one course all the way, but longer. The shortest track is the great circle, which on a Mercator chart bows towards the nearer pole.
A great circle is cut by a plane through the Earth's centre; its shorter arc is the shortest track between two points. A rhumb line crosses every meridian at the same angle: one course all the way, straight on a Mercator chart, but longer.
Yokohama to Victoria: great circle 4,104 M, rhumb line 4,374 M.
On a Mercator chart the great circle bows towards the nearer pole, an S across the equator. The saving grows with latitude and east–west distance: 214 miles Cape Town to Fremantle, 10 Canaries to St Lucia, none along a meridian or the equator, which are great circles themselves. Other parallels are small circles: even a due-east leg is not the shortest.
In the plotter, leg type GC follows it; RL holds one course.
2 · Distance and initial course
A triangle on the globe
How do I calculate the great circle distance between two positions?
Solve the spherical triangle of pole, departure and destination: cos d = sin L1 × sin L2 + cos L1 × cos L2 × cos DLo, contrary-name latitudes negative. Each degree of d is 60 miles.
The pole, departure A and destination B form a spherical triangle: two sides are 90° minus each latitude, the angle at the pole is the difference of longitude (DLo), the third side the distance and the angle at A the initial course.
Distance
cos d = sin L1 × sin L2 + cos L1 × cos L2 × cos DLo. Each degree of d is 60 miles.
Contrary-name latitudes go in negative. For DLo subtract longitudes of the same name, add east to west; across the date line add both distances from 180°. Initial course: tan C = sin DLo ÷ (cos L1 × tan L2 − sin L1 × cos DLo), from north towards your side; if negative, add 180°.
45°S 080°E → 45°S 140°E
DLo 60°E, both latitudes south (−45°) cos d = 0.5 + 0.5 × cos 60° = 0.75 d = 41.41° × 60 = 2,485 M tan C = −2.449: C = 180° − 67.8° = 112°T
Along the parallel: 2,546 M
3 · The vertex
Where the track turns back
Why does my course keep changing when I sail a great circle?
A great circle cuts each meridian at a different angle, so the course keeps changing; it runs due east or west at the vertex, the point nearest the pole. Hold the first course and you sail a rhumb line past your port.
A great circle cuts each meridian at a different angle, so the course changes all the way. The vertex is the point nearest the pole: there the track runs due east or west, then turns back towards the equator. Hold the first course and you sail a rhumb line past your port.
112°T at the start, 090°T at the vertex, 068°T on arrival.
Vertex
cos Lv = cos L1 × sin C; to its meridian, cos DLo = tan L1 ÷ tan Lv.
The vertex may lie off the passage: beyond the destination if you arrive still closing the pole, behind you if you start heading away. The track crosses the equator 90° of longitude from the vertex; at any latitude L on it, sin C = cos Lv ÷ cos L. Blown off the track, start a new great circle from your fix.
4 · Gnomonic chart and waypoints
Putting the curve on your chart
How do I plot a great circle route on a Mercator chart?
Rule it as a straight line on a gnomonic chart, read its latitude on each meridian 5° or 10° apart, plot those points on the Mercator chart and join them with short rhumb lines.
On a gnomonic chart every great circle is a straight line. Rule it, but measure neither course nor distance there: both are distorted. Read its latitude on each meridian, 5° or 10° apart, plot the points on the Mercator chart and join them with short rhumb lines.
Or compute them from the vertex: tan L = tan Lv × cos (DLo from the vertex). Along 45°S:
Meridian
Latitude
080°E, 140°E
45°00′S
090°E, 130°E
47°20′S
100°E, 120°E
48°40′S
110°E (vertex)
49°06′S
Legs 10° apart add only a mile or two; space them closer in high latitudes, where the course turns fastest.
5 · Composite sailing
A latitude you will not cross
How do I keep a great circle route out of the ice in high latitudes?
Sail a composite track: a great circle whose vertex lies on your chosen limiting latitude, along that parallel, then a second great circle to the destination. It is the shortest track that keeps to the limit.
In high latitudes the great circle can lead into ice: from Bluff to Cape Horn its vertex is near 70°S. Choose a limiting latitude and sail a composite track: a great circle whose vertex lies on the limit, the parallel itself, then a second great circle to the destination: the shortest track that keeps to the limit. Each leg is solved from its vertex on the limit.
Bluff to Cape Horn, limit 58°S: 4,297 M; the direct great circle 4,040 M.
First leg: Bluff, 46°36′S, to 58°S
sin C = cos 58° ÷ cos 46.6° → 130°T cos DLo = tan 46.6° ÷ tan 58° → 48°38′ cos d = sin 46.6° ÷ sin 58° → 1,863 M
Along 58°S: 3,212′ × cos 58° = 1,702 M
Total 1,863 + 1,702 + 732 = 4,297 M
In the plotter: two GC legs joined by an RL leg along the parallel.
6 · Rhumb-line sailings
Plane, parallel and Mercator
How do I work out my DR position from a day's run by calculation?
Use plane sailing: dLat = d × cos C, departure = d × sin C, and DLo = departure ÷ cos mean latitude. Over a long rhumb line the mean latitude no longer gives the true departure: use Mercator sailing.
Rhumb lines are worked by three sailings. Departure is the east–west distance in miles: away from the equator a minute of longitude is less than a mile.
Sailing
Use it for
Formulas
Plane
A day's run, a few hundred miles
dLat = d × cos C; dep = d × sin C; DLo = dep ÷ cos mean lat
Parallel
Due east or west
dep = DLo × cos lat
Mercator
A long rhumb line
tan C = DLo ÷ DMP; d = dLat ÷ cos C
Plane sailing: a day's run
From 20°00′N 030°00′W: 240°T, 120 M dLat = 120 × cos 60° = 60′ S dep = 120 × sin 60° = 103.9′ W
DLo = 103.9 ÷ cos 19.5° = 110′ W
DR 19°00′N 031°50′W
DMP is the difference of meridional parts: subtract for same-name latitudes, add across the equator. Over a long leg the mean latitude no longer gives the true departure: use Mercator sailing.
7 · Routing
More miles, fewer days
Is the great circle always the fastest route for a sailing yacht?
No: the shortest route is not always the fastest or safest, so a yacht routes for wind, current and season first. From the Canaries, for example, go south to the steady NE trade, then west.
The shortest route is not always the fastest or safest: a yacht routes for wind, current and season first.
Honolulu to San Francisco in summer: north round the Pacific High, then east in the westerlies.
Trade winds: from the Canaries go south to the steady NE trade, then west.
Highs: round the calm centre, on the side with a fair wind.
High latitudes: stay on the equator side of the great circle, clear of ice, fog and gales.
Monsoons: pick the season that puts the wind on your quarter.
Practice questions
Questions from this topic, as the app asks them. Open one to see the answer.
Plotter: Cape Town → Fremantle
Chart view: Mercator
GC track bows south to about 44½°S
GC 4,686 M · rhumb line 4,900 M
Before the Indian Ocean crossing a crew member looks at the plotter and asks why the great-circle track, which curves far south of the straight line on the chart, is the shorter one. What do you tell them?
Its plane passes through the Earth's centre — the shortest way on a sphere
The plotter allows for the east-going current in the forties
It follows the line along which magnetic variation stays the same
It keeps one course, so no miles are lost in turns
Show the answer
A. Its plane passes through the Earth's centre — the shortest way on a sphere
A great circle is cut by a plane through the Earth's centre; its shorter arc is the shortest route between two points on the globe.
All 100 questions of this topic are in the app: practice by topic with the reason after every answer, your mistakes brought back, and a timed mock exam. Free, no sign-up, in the browser or in Telegram.
