Calculation of distances between points on their coordinates on the plane is elementary, at the Earth's surface - a bit more complicated: we will measure the distance and initial bearing between points without the projective transformation. To start sort out the terminology.
The initial azimuth - the azimuth, which took at the beginning of the movement from point A, following the great circle on the shortest distance to point B, the endpoint is the point B. When moving from point A to point B through the great circle azimuth from the current position to the endpoint B is constantly changing. The initial azimuth is different from the constant, following which, the azimuth of the current point in the final does not change, but the route is not the shortest distance between two points.
Through any, two points on the surface of a sphere, if they are not directly opposite each other (i.e. are not antipodes) can be a unique great circle. Two points separated the big circle into two arcs. The length of the short arc - the shortest distance between two points. Between two points-antipodes can be an infinite number of great circles, but the distance between them will be the same on any disk and equal to half the circumference of a circle, or π * R, where R - radius of the sphere.
On the plane (in a rectangular coordinate system), large circles, and fragments thereof, as mentioned above, represent the arc in all projections, except the gnomonic, where the large circles - direct line. In practice, this means that aircraft and other air transport route always uses the minimum distance between points for fuel economy, that is carried on a flight distance of a great circle on the plane, it looks like an arc.
The shape of the Earth can be described as a sphere, so the equation for calculating distances on a large circle are important for calculating the shortest distance between two points on Earth's surface and are often used in navigation.
Calculating the distance by this method is more effective and in many cases more accurate than the calculation for its projected coordinates (in rectangular coordinates), because, firstly, there's no need to convert geographic coordinates to a rectangular coordinate system (to the projection transformation), and Second, many of the projections, if properly selected, can lead to significant distortions of lengths due to the peculiarities of projection distortion.
It is known that more accurately describes the shape of the Earth is not a sphere, and ellipsoid, but in this article the calculation of distances is on the field, is used to calculate the sphere of radius of the earth in meters - 6372795 meters, which can lead to a calculation error distances of about 0.5%.