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Orthodromy - Loxodromy - Navigation - Course

The terms above are the core subjects covered on this website and so I thought they required a little more explanation than just a blunt definition in the glossary.

An orthodromic or great-circle route on the Earth's surface is the shortest possible real way between any two points.

The shortest way? A straight line, some will say!
And there I say: NO! We're not drilling our way through the Earth. I said 'real' way. Because the Earth is a sphere (although not exactly spherical), the way is in fact an arc (also called 'great-circle arc') the diameter of which is that of the Earth.
To better understand, take a globe and set a thread between two points: this is the orthodromic route (or great-circle route). It's also the route a bird would fly to go from A to B.

Caution! Great-circle (or orthodromic) distances aren't straight lines! If you draw a straight line on a map, what you get is a loxodromic route, not a great-circle route. Here's why: great-circle distances are calculated on a sphere. Now, on a classic map (cylindric projection), surfaces and proportions are distorted and wrong, especially as one gets nearer to the poles. For example, let's go back to our globe and thread: take the thread and make it go through a few big cities on a sloping plane. And on a map, draw a line between the same cities. As a result, you will see that both lines are far from 'identical'!

So what exactly is loxodromy? Simply a line on a map i.e. a curve that goes from A to B following a fixed compass heading. In other words, you follow the direction given by the compass from the beginning until the end of the route. The curve is actually longer than the great-circle distance because in this case map projections aren't taken into account.

So now you know that to go as 'fast' as possible from A to B one should not follow a fixed course: the course varies in a great-circle route but stays the same all the way through in a loxodromic route. GPS systems can calculate the course in real-time because they always know your exact position.

To finish, here's the formula used to calculate a great-circle distance in kilometers from A to B:

Ortho(A,B)=6371 x acos[cos(LatA) x cos(LatB) x cos(LongB-LongA)+sin(LatA) x sin(LatB)]

Where 6371 is the Earth's radius in kilometers.

6371 isn't an exact value: The Earth isn't a perfect sphere but rather a spheroid. 6371 is the mean radius: (more info here (in French)).

I won't try to demonstrate the formula myself as others did it very well before me (see the links section).

Want to ask a question? please visit the

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