Widening

The Abweitung (engl. Departure ) denotes the length of a width of the circular arc between two points of the same latitude on the earth's surface . The expansion is greatest at the equator with about 111 km with a length difference of 1 ° and decreases towards the poles where it has the value zero. The expansion is - apart from the equator - greater than the shortest distance on the earth's surface between the two points, since the equator is the only circle of latitude that is a great circle . The expansion differs significantly from the distance between two points along a meridian, because this (on the sphere) only depends on the difference in width, not on the width itself or the length. ${\displaystyle \,\phi }$

In nautical science , the widening is the leg in the course triangle that coincides with the parallel circle .

The definition of the widening is often restricted to the distance along a circle of latitude between two meridians that are exactly 1 ° apart.

calculation

On a spherical reference surface, the length of the arc of the circle between two points of latitude and longitude and with and the circumference of the circle of latitude is calculated from: ${\displaystyle \,\phi }$ ${\displaystyle \,\lambda _{1}}$ ${\displaystyle \,\lambda _{2}}$ ${\displaystyle \Delta \lambda =|\lambda _{2}-\lambda _{1}|}$ ${\displaystyle U_{\phi }}$

${\displaystyle \mathrm {Abweitung} ={\frac {\Delta \lambda }{360^{\circ }}}\cdot U_{\phi }.}$

The circumference of a parallel depends on the geographical latitude and the earth's radius or circumference : ${\displaystyle R}$ ${\displaystyle U}$

${\displaystyle U_{\phi }=2\pi R\cos \phi =U\cos \phi .}$

For the widening on a spherical reference surface this results in:

${\displaystyle \mathrm {Abweitung} ={\frac {\Delta \lambda \cdot U}{360^{\circ }}}\cos(\phi ).}$

If a reference ellipsoid is used as a reference surface for more precise calculations , the circumference of a circle of latitude cannot be calculated as above. Instead, the radius of curvature of the latitude , i.e. H. the normal radius of curvature of a geodetic line perpendicular to the meridian, used: ${\displaystyle N(\phi )}$ ${\displaystyle \,\phi }$

${\displaystyle \mathrm {Abweitung} ={\frac {\Delta \lambda }{360^{\circ }}}\cdot 2\pi N(\phi )\cdot \cos(\phi ).}$

values

The following table shows the deviation as a function of the geographical latitude for two reference ellipsoids. For comparison, the last column shows the length of the meridian arc between latitude and latitude , the dependence of which on latitude is small. ${\displaystyle \,\phi }$ ${\displaystyle \,\phi -0{,}5^{\circ }}$ ${\displaystyle \,\phi +0{,}5^{\circ }}$

Widening					Meridian arc length
	Bessel ellipsoid			WGS84
width	1 ° [km]	1 ′ [m]	1 ″ [m]	1 ° [km]	1 ° [km]
0 °	111,307	1855	30.9	111,319	110.574
10 °	109.627	1827	30.5	109.639	110.608
20 °	104.635	1744	29.1	104.647	110.704
30 °	96.475	1608	26.8	96.486	110.852
40 °	85.384	1423	23.7	85.394	111.035
45 °	78.837	1314	21.9	78.847	111.132
50 °	71.687	1195	19.9	71.696	111.229
60 °	55.793	930	15.5	55,800	111.412
70 °	38.182	636	10.6	38.187	111,562
80 °	19,391	323	5.4	19,393	111.660
90 °	0, 000	0	0, 0	0, 000	111,694

Geographical coordinates that are accurate to the second are therefore accurate to within 20 meters in Central Europe (at around 49 ° latitude). In terms of width, on the other hand, they have an accuracy of about 30 meters regardless of location. Position to the meter requires at least the second decimal place of the decimal seconds.

Individual evidence

1. ↑ deviation (declination) . In: Meyers Großes Konversations-Lexikon. Volume 1. Leipzig 1905, p. 66. ( zeno.org )
2. ↑ for example: expansion . In: Geoinformatics Service. Dictionary. geoinformatik.uni-rostock.de
3. ↑ Quotation after widening . In: Lexicon of Cartography and Geomatics. Spektrum Akademischer Verlag, Wissenschaft-online.de (only for 1 °, the rest calculated)