# Precision of coordinates

## Definition

1 arc minute along a meridian or along the Equator (over the WGS84 reference geoid) equals 1 852 m.

1′ = 1 852 m

## Precision of latitudes

DD.dddddd° Decimal places DD°MM′SS.sss″ Metres
0 111 120
0.1° 1 0°06′ 11 112
0.01° 2 0°00′36″ 1 111.2
0.001° 3 0°00′03.6" 111.12
0.000 1° 4 0°00′00,36″ 11.112
0.000 01° 5 0°00′00.04″ 1.1112
0.000 001° 6 0°00′00.004″ 0.11112
MM.m′ DD.dddddd° Metres
10′ 0.166° 18 520
1′ 0.016 6° 1 852
0.1′ 0.001 66° 185.2
0.01′ 0.000 166° 18.52
0.001′ 0.000 016 6° 1.852
0.000 1′ 0.000 001 66° 0.185 2
SS.sss″ DD.dddddd° Metres
10″ 0.027 7° 308.6
1″ 0.002 77° 30.86
0.1″ 0.000 277° 3.086
0.01″ 0.000 027 7° 0.308 6
0.001″ 0.000 002 77° 0.030 86

## Precision of longitudes

Latitude Cosinus
0° (equator) 1.00
10° 0.98
20° 0.94
30° 0.86
40° 0.77
50° 0.64
60° 0.50
70° 0.34
80° 0.17

The precision of longitudes is dependent on the latitude : the higher the latitude, the closer the meridians are to each other. The value in meters is to be multiplied by the cosine of the latitude (except along the equator). So, for example, in Germany (50 ° N) the meridians are about 2/3 as large as on the equator and the accuracy is correspondingly higher.

Precision of given longitude Distance along a parallel depending on latitude
DD.dddddd° Decimal places DD°MM′SS.sss″ 0° (equator) 30° 45° 60° 75°
0 111 120 m 96 233 m 78 574 m 55 560 m 28 760 m
0.1° 1 0°06′ 11 112 m 9 623 m 7 857 m 5 556 m 2 876 m
0.01° 2 0°00′36″ 1 111.2 m 962 m 785 m 555 m 288 m
0.001° 3 0°00′03.6″ 111.12 m 96 m 78 m 55 m 29 m
0.000 1° 4 0°00′00.36″ 11.112 m 9.6 m 7.8 m 5.5 m 2.9 m
0.000 01° 5 0°00′00.036″ 1.111 2 m 0.96 m 0.78 m 0.55 m 0.29 m
0.000 001° 6 0°00′00.003 6″ 0.111 12 m 0.096 m 0.078 m 0.055 m 0.028 m

## Distance

The loxodromic distance between two points A and B (on the WGS84 reference geoid) may be approximated by twice the quadratic average of the two distances below if the difference of latitudes is not too important (less than about 10°):

latitudinal distance along any meridian (exact)
latitudinal distance (in metres) = (decimal latitude A - decimal latitude B) * 111 120 m
longitudinal distance along an average parallel (approximation; the highest the difference of the two latitudes, the lowest is the precision)
longitudinal distance (in metres) ≈ (decimal longitude A - decimal longitude B) * cos(average latitude) * 111 120m

In practice, most objects in OSM (including the largest ones such as coastlines and land boundaries of countries) are traced with small segments whose two end points have very near latitudes whose difference is much below 1°; if this is not the case the polygons should be improved to add missing intermediate points if arcs are not traced along a parallel or meridian (this should be done for roads).

The effective distance on highways/railways/waterways cannot be computed exactly this way, because of approximation of curves by polylines, and because of additional differences of altitude (and imprecision on the terrain data model) and the impossibility to follow exactly an averaged theoretical curve.

## Conversion to decimal

Accuracy DD.dddddd°
decimal places
10 m 4
1 m 5
0.1 m 6

If coordinates in [DD°MM′SS.sss″] are converted to [DD.dddddd°] or vice versa, then sometimes a higher number of decimal places are created, which simulate a greater accuracy than what actually exists.

For conversions, it makes sense to round out the results to reflect the original accuracy.

Even better we could add a separate key for accuracy: accuracy=##.# (in metres)

## Gauss-Krüger conversion in WGS 84

Wenn Gauß-Krüger-Koordinaten in Koordinaten des World Geodetic System 1984 (WGS 84) umgerechnet werden oder umgekehrt, dann entstehen je nach verwendeter Formel größere Ungenauigkeiten.

Auch hier können dabei mehr oder weniger Nachkommastellen entstehen, die aber nichts über die Genauigkeit aussagen.

## Display and measurement accuracy

Achtung: eine vielstellige Anzeige bedeutet nicht eine genaue Messung.

Wenn ein Gerät eine 8-stellige Anzeige für dezimale Koordinaten verwendet, also eine Messung von 1 cm anzeigen könnte, so funktioniert das nur, wenn auch das Gerät selbst eine Messgenauigkeit von 1 cm erreicht. Wenn das Gerät aber beispielsweise eine Messgenauigkeit von 10 m aufweist, so dürfen in der 8-stelligen Anzeige die letzten 3 Stellen der Anzeige nicht berücksichtigt werden.

sinnvolle Anzeigegenauigkeit

Die letzte Stelle der Anzeige entspricht - und erfordert! - folgende Messgenauigkeit:
(bezogen auf den Grosskreis)

Measurement accuracy Display accuracy
DD.dddddd°
Nautic miles Metres
0.0006 nmi 1.111 m ##.#####°
0.006 nmi 11.11 m ##.####°
0.06 nmi 111.1 m ##.###°
0.6 nmi 1 111 m ##.##°

Display accuracy
DD°MM′SS.sss″
Measurement accuracy
Nautic miles Metres
##°##′##.#″ 0.0016 nmi 3.1 m
##°##′##″ 0.016 nmi 31 m
##°##.#′ 0.1 nmi 185 m
##°##′ 1 nmi 1 852 m