Rate function

Decimal degrees (DD) express latitude and longitude geographic coordinates as decimal fractions and are used in many geographic information systems (GIS), web mapping applications such as Google Maps, and GPS devices. Decimal degrees are an alternative to using degrees, minutes, and seconds (DMS). As with latitude and longitude, the values are bounded by ±90° and ±180° respectively.

Positive latitudes are north of the equator, negative latitudes are south of the equator. Positive longitudes are east of Prime Meridian, negative longitudes are west of the Prime Meridian. Latitude and longitude are usually expressed in that sequence, latitude before longitude.

Accuracy

The radius of the semi-major axis of the Earth at the equator is 6,378,160.0 meters resulting in a circumference of 40,075,161.2 meters. The equator is divided into 360 degrees of longitude, so each degree at the equator represents 111,319.9 metres or approximately 111.32 km. As one moves away from the equator towards a pole, however, one degree of longitude is multiplied by the cosine of the latitude, decreasing the distance, approaching zero at the pole. The number of decimal places required for a particular accuracy at the equator is:

Accuracy versus decimal places

decimal places	degrees	N/S or E/W at equator	E/W at 23N/S	E/W at 45N/S	E/W at 67N/S
0	1.0	111.32 km	102.47 km	78.71 km	43.496 km
1	0.1	11.132 km	10.247 km	7.871 km	4.3496 km
2	0.01	1.1132 km	1.0247 km	.7871 km	.43496 km
3	0.001	111.32 m	102.47 m	78.71 m	43.496 m
4	0.0001	11.132 m	10.247 m	7.871 m	4.3496 m
5	0.00001	1.1132 m	1.0247 m	.7871 m	.43496 m
6	0.000001	111.32 mm	102.47 mm	78.71 mm	43.496 mm
7	0.0000001	11.132 mm	10.247 mm	7.871 mm	4.3496 mm
8	0.00000001	1.1132 mm	1.0247 mm	.7871 mm	.43496 mm

A value in decimal degrees to an accuracy of 4 decimal places is accurate to 11.132 meters (± 5.566 m) at the equator. A value in decimal degrees to 5 decimal places is accurate to 1.1132 meter at the equator. Elevation also introduces a small error. At 6,378 elevation, the radius and surface distance is increased by 0.001 or 0.1%. Because the earth is not flat, the accuracy of the longitude part of the coordinates increases the further from the equator you get. The accuracy of the latitude part does not increase so much, more strictly however, a meridian arc length per 1 second depends on latitude at point concerned. The discrepancy of 1 second meridian arc length between equator and pole is about 0.3 metres because the earth is an oblate spheroid.

Example

A DMS value is converted to decimal degrees using the formula:

$\mathrm{D}\mathrm{D}=\mathrm{D}+\frac{\mathrm{M}}{60}+\frac{\mathrm{S}}{3600}$

For instance, the decimal degree representation for

38° 53′ 23″ N, 77° 00′ 32″ W

(the location of the United States Capitol) is

38.889722°, -77.008889°

In most systems, such as Google Maps, the degree symbols are omitted, reducing the representation to

38.889722, -77.008889

To calculate the D, M and S components, the following formulas can be used:

$\mathrm{D}\mathrm{D}=\{\begin{array}{ll}\mathrm{D}& =\mathrm{trunc}(\mathrm{D}\mathrm{D})\\ \mathrm{M}& =\mathrm{trunc}(|\mathrm{D}\mathrm{D}|*60)mod60\\ \mathrm{S}& =\left(\right|\mathrm{D}\mathrm{D}|*3600)mod60\end{array}$

where |DD| is the absolute value of DD, trunc is the truncation function, and mod is the modulo operator. Note that with this formula, only S may have a fractional value.

Pseudo code, using numbers from above:
$min=int(mod(38.889722*60,60)); $sec=round(mod(38.889722*3600,60)) => min:53, sec:23

References

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