# Beeline The church towers are about 17.5 km as the crow flies and separated by a state border: in the
front right the Catholic Church of St. James in Schutterwald (Germany), in the back left the minster in Strasbourg (France)

As the crow flies , one describes the shortest distance between two points in the landscape by direct air if the two points are in line of sight. In this case, the beeline is a distance (which may also overcome major differences in altitude in the terrain, for example in the mountains). Is the line of sight through an obstacle, e.g. If, for example, a building or a mountain is interrupted, the straight line corresponds to the distance between the two points if the obstacle were not there.

At greater distances, the beeline does not take into account the contour of the terrain - that is, elevations, valleys and differences in height - but does include the spherical shape of the earth. In this case the beeline runs “horizontally” and follows the curvature of the earth; From a mathematical point of view, the beeline here corresponds to an arc of a circle that lies on a great circle around the center of the earth (compare spherical trigonometry ). When such routes are projected onto flat maps, there are generally no longer any straight lines , but curves that still represent the shortest distance between two points. For example, the straight line between New York and Berlin runs through Scotland. In geometry and navigation , one speaks more precisely of the orthodrome instead of a straight line.

When sailing, preference is given to the Loxodrome instead of the Orthodrome: the Loxodrome is characterized by the fact that the bearing angle to the target does not change.

A map projection in which great circles (and thus the air lines between two points) are always depicted as straight lines is the gnomonic projection .

## Calculation for the globe

The earth can be viewed as a sphere to a good approximation. To simplify matters, the radius of the sphere can be assumed to be one. The Cartesian coordinates are calculated from the latitude and longitude of a point - with the -axis in the direction of the earth's axis - with the help of the trigonometric functions sine and cosine : ${\ displaystyle \ varphi}$ ${\ displaystyle \ lambda}$ ${\ displaystyle P}$ ${\ displaystyle (x, y, z)}$ ${\ displaystyle z}$ ${\ displaystyle (x, y, z) = (\ cos (\ varphi) \ cdot \ sin (\ lambda), \ cos (\ varphi) \ cdot \ cos (\ lambda), \ sin (\ varphi))}$ Another point on the globe has the same coordinates ${\ displaystyle P '}$ ${\ displaystyle (x ', y', z ') = (\ cos (\ varphi') \ cdot \ sin (\ lambda '), \ cos (\ varphi') \ cdot \ cos (\ lambda '), \ sin (\ varphi '))}$ First of all, the Euclidean distance between the two points in three-dimensional space can be calculated with the Pythagorean theorem (this is not the straight line, but the length of the distance that leads through the globe):

${\ displaystyle d (P, P ') = {\ sqrt {(x-x') ^ {2} + (y-y ') ^ {2} + (z-z') ^ {2}}}}$ At every point of this line there is a perpendicular that is perpendicular to the earth's surface (there is a point of the orthodrome) and consequently goes through the center of the earth. All points of such a perpendicular have the same geographic coordinates , but a different radius (distance from the center of the earth). If the radius of the earth is used as the radius , the geographic coordinates for each point of the orthodrome can be calculated. ${\ displaystyle R}$ The opening angle can now be calculated from the distance and the radius of the earth : ${\ displaystyle d}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ sin (\ omega / 2) = {\ frac {d / 2} {R}} \ qquad \ Rightarrow \ omega = 2 \ cdot \ arcsin \ left ({\ frac {d / 2} {R} } \ right)}$ Alternatively, the opening angle can be calculated with the scalar product:

${\ displaystyle \ cos (\ omega) = {\ frac {{\ vec {a}} \ cdot {\ vec {b}}} {| {\ vec {a}} || {\ vec {b}} | }} = {\ frac {x * x '+ y * y' + z * z '} {{\ sqrt {x ^ {2} + y ^ {2} + z ^ {2}}} {\ sqrt { x '^ {2} + y' ^ {2} + z '^ {2}}}}}}$ , that simplifies the equation.${\ displaystyle | {\ vec {a}} | = | {\ vec {b}} | = 1}$ ${\ displaystyle \ Rightarrow \ omega = \ arccos \ left (x * x '+ y * y' + z * z '\ right)}$ Alternatively, the opening angle can also be calculated directly from the geographic coordinates:

${\ displaystyle \ cos (\ omega) = \ cos (\ varphi) \ cdot \ cos (\ varphi ') \ cdot \ cos (\ lambda - \ lambda') + \ sin (\ varphi) \ cdot \ sin (\ varphi ') \ qquad \ Rightarrow \ omega = \ arccos \, (\ dots)}$ The straight line you are looking for is the length of the arc and the opening angle in radians multiplied by the radius of the earth: ${\ displaystyle L}$ ${\ displaystyle L = \ omega \ cdot R}$ In the same way, the apparent distance in radians between two stars with a given declination and right ascension can be calculated.