Simply stated, the shortest possible route between two points on the Earth's surface follows a great circle. It should also be stated that following the shortest distance to your destination does not necessarily mean you will spend the least time getting there if, for example, you're on foot and the Alps happen to be in the way, or you're sailing and the doldrums happen to lie along your path. Outside of these and other similar inconveniences, a great circle route is ideal.
You can find the great circle route between any two points on a globe by pulling a string tightly between both hands and pressing the string down on the globe while also keeping it taut. If you manage to do this so that the string touches the two points, you have found the route in-between that uses the least string and is therefore the shortest distance. Congratulations.
Unfortunately, globes do not fit handily into small spaces such as control rooms or cockpits and it is more likely that you will find a map drawn on a flat sheet instead. We will return to this problem later, but firstly, a definition is in order.
A Great Circle is the largest circle that can be drawn on a sphere1. To see this, picture the trace of an imaginary sphere as it is held above a sheet of imaginary paper and slowly lowered so that it passes through the sheet. When the bottom of the sphere first touches the sheet a dot will appear. As it continues, a circle will appear on the sheet that will become larger and larger until the sphere is exactly halfway through. Then the circle will diminish down to a point again and disappear. So, the largest possible circle occurs when half of the sphere is above the sheet and half is below.
The equator is one such great circle since it divides the Northern and Southern hemispheres. It is the only line of latitude that is a great circle, and the only one on which a due east or west compass bearing2 ensures the shortest course. In contrast, all lines of longitude, also called meridians, are great circles and maintaining a compass bearing of north or south always ensures the shortest course. In any but these special cases, the compass bearing along a great circle route will constantly change.
Constant Bearing Route
Early navigators would have preferred a course on which the compass bearing remained constant. While following a course of constant bearing is harder on the legs because it will generally be longer, it is easier on the brain because it requires no computation. On Mercator projections3, which were common among early maps made by Gerardus Mercator4, and which also required no computation5, a line of constant bearing is any straight line drawn on the map - this was in fact the goal of Gerardus. On the Earth, such a path will appear as a spiral. On the other hand, great circle routes that follow neither the equator nor any meridian always appear curved on Mercator projections6.
A Great Circle Map
If you must use a map to navigate a great circle route, then it is best to use one made for the purpose. The gnomonic projection makes just such a one. The gnomonic projection is accomplished by holding a flat sheet of paper against a lighted globe so that the centre of the sheet touches the globe at the point you wish to depart from. Then any straight line drawn through this point on the projection represents a great circle. Handy, but a new map will be needed whenever you wish to change your point of departure.
A Modification to the Great Circle
As stated at the outset, things can get in the way of your great circle route. The following is taken from Bowditch, American Practical Navigator, 1938, page 105:
It frequently happens when a great circle route is laid down that it is found to lead across the land, or carry the vessel into a region of dangerous navigation or extreme cold which it is expedient to avoid; in such a case a certain parallel should be fixed upon as the limit of latitude, and a route laid down such that a great circle is followed as far as the limiting parallel, then the parallel itself, and finally another great circle to the port of destination. Such a modification of the great circle method is called composite sailing.
Using the Stars
The Polynesians were known to navigate by the stars and follow great circle routes. To understand this, first suppose that you are somewhere in the northern hemisphere and you wish to travel to the North Pole. It happens that Polaris, also called the Pole Star, hangs out above the North Pole. So, if you set your course straight for this beacon, you will be travelling due north along a line of longitude and your route is a great circle. If you wish to travel to some place other than the North Pole, well then, you have a problem because all other stars in the sky appear to rotate about Polaris and won't stay put. However, if you were able to stop the universe for just a moment, then at that precise moment some star would be stationary above your chosen destination. You could set your course straight for that star and, here's glory for you, you will be on a great circle route.
Suppose that you, as an ancient or at least antique Polynesian, while unable to stop the universe, are instead very observant and you have kept track of which stars pass above which islands. You notice that the same stars pass over the same islands, but appear somewhat earlier each evening. If your sense of time is reasonably acute, as it very probably is7, then you have a very good idea when a particular star is above a particular island based on when that particular star appeared over your horizon and how far away you reckon that island to be. Now, putting this all together, say that you know it is about two hours after sunset and that a particular star is above Hawaii. If you set a course for that star, you are on a great circle route to the Big Island. Well - at least for a little while, but now it's three hours after sunset and you know which star is the next above Hawaii, and so on. You can adapt this method to modern times by calling or emailing someone at your destination and asking which star is directly overhead and then setting your course for it. Unfortunately, cloudy skies will foil this method.
If you enjoy this kind of thing, then you may want to read Finding Your Way Without Map or Compass by Harold Gatty. Most libraries have it - first, call the library some evening and ask them which star is overhead at the moment, then you can go straight there.