Beeline

The title of this article is ambiguous. For a straight line when designing printed circuit boards, see PCB unbundling .

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.

Web links

Wiktionary: beeline  - explanations of meanings, word origins, synonyms, translations

• Interactive calculation of the straight line distance between two locations with visualization of the great circle arc on a map:
  • luftlinie.org (Stephan Georg)
  • Distance (Martin Kompf)

Individual evidence

1. ↑ Distance New-York → Berlin (luftlinie.org)
2. ↑ Definition in the Duden
3. ↑ ^{a b} Distance calculation at Kompf.de