# 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.

The shortest path on the earth's surface between point A and B is an orthodrome.

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.