As the crow flies
When a straight line is defined as the shortest distance between two points in the landscape beyond the direct air through a distance. Air lines are always parallel to the surface.
For longer distances, the spherical shape of the earth is taken into account. In spherical trigonometry one then moves along arcs of a circle whose center is the center of the earth (so-called great circles ). When projecting such distances on flat cards generally are no longer straight lines, but curves, but still represent the shortest distance between two points. In geometry and navigation used is therefore precisely the geodesic instead of a straight line.
A map projection in which great circles (and thus the " air lines " between two points ) are always represented as a straight line, is the Gnomonic projection.
Mathematical calculation for the globe
Earth may be considered a good approximation as a ball. To simplify the radius of the sphere may be assumed to be one. From the latitude and the longitude of a point, the Cartesian coordinates are calculated with the- axis in the direction of Earth's axis using the trigonometric functions sine and cosine.
Another point on the globe has similarly the coordinates
First, can the Pythagorean Theorem
The distance in three-dimensional space of the two points are calculated. This is not a straight line, but the length of the path that leads through the globe. For each point of this path there is a solder that is perpendicular to the earth's surface ( there is a point of the geodesic ) and therefore passes through the center of the earth. All points of the solder have the same geographic coordinates, but a different r. Thus, for each item on the geodesic calculate the geographic coordinates.
The sine of the half apex angle of W is the ratio of half the distance and the radius of the Earth. The length of the arc, the required straight line is the opening angle in radians multiplied by the radius of the earth.
The following applies:
In the same way also the apparent distance in radians two stars are calculated with a given declination and right ascension.