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Geographic coordinate conversion

Geographic coordinates consist of latitude and longitude. There are many ways of writing coordinates, and converting between the different ways is non-obvious but also quite trivial.

Table of contents
1 Ways of Writing Coordinates
2 Basic Forms
3 Components of a Typical Coordinate
4 Putting it all Together

Ways of Writing Coordinates

All of the following are valid and acceptable ways to write geographic coordinates:

Basic Forms

There are three basic forms of a coordinate.

  1. Coordinate containing degrees (integer), minutes (integer), and seconds (integer, or real number).
  2. Coordinate containing degrees (integer) and minutes (real number).
  3. Coordinate containing only degrees (real number).

All forms of coordinates are capable of representing the same amount of data and the same precision. Depending on which type of coordinate you are provided with, and which type you would like to work with, you may have to do some conversion.

Components of a Typical Coordinate

In it's most simple form a coordinate is just a number of degrees. The tricky part comes in when you need to differentiate North/South latitude or West/East longitude, or make the number more digestable by writing it with minutes and seconds instead of as a decimal number.

Degrees

The degrees portion of the coordinate is always going to be the easiest to figure out. The degrees is always the left-most whole number. For example:

40:26:46N         40
W87°43'41         87

A sphere is divided into 360 degrees. The number space is divided into two halves, East and West in the case of longitude and North and South in the case of latitude. The maximum ranges are as follows:

Longitude
 180 W   = -180
 180 E   =  180

Latitude 90 N = 90 90 S = -90

Technically you could have latitudes greater than 90 or less than -90, but this is an ambiguous case, since there would be an equivalent coordinate with an inverse longitude.

The minimal case is that you have only degrees:

 40.446111  or
 40.446111N

Minutes

Minutes are an optional component, as is implied by the minimal case of degrees. If there is no minutes component, the degrees component contains the entire precision of the coordinate and there must not be a seconds component. Minutes are actually the numerator component of a fraction with denominator 60 of one degree.

With the same examples as above:

40:26:46N         26
W87°43'41         43

In the first case, the number of minutes is 26.

To convert, 26 minutes is equal to degrees.

Seconds

Seconds are also an optional component, and can only exist if the minutes component also exists. Seconds are the numerator component of a fraction with denominator 60 of one minute.

40:26:46N         46
W87°43'41         41

In the second case, the number of minutes is 41.

To convert, 41 seconds is equal to minutes.

Putting it all Together

Conversion from DMS to Decimal

Given a DMS coordinate such as W87°43'41", it's trivial to convert it to a number of decimal degrees using the following method:

Conversion from Decimal to DMS

Given a decimal longitudinal coordinate such as -87.728055 it is trivial to convert it to DMS form. It will be necessary to know whether it is a latitudinal or longitudinal coordinate in order to fully convert it. The method is as follows:

  Type   Dir.   Sign    Test
  Lat.   N      +       > 0
  Lat.   S      -       < 0
  Long.  E      +       > 0
  Long.  W      -       < 0

A coordinate with at 0°0'0" latitude or longitude is neither North nor South, East nor West. It is simply zero latitude or zero longitude.

Programmatical Conversion

The most common programmatical use of these processes is to display a coordinate to an end user in the more common DMS form instead of decimal form. Some example code in the PHP programming language to do this is:

function pretty_coord($coord) {
  return sprintf("%0.0f %2.3f",
                 floor(abs($coord)),
                 60*(abs($coord)-floor(abs($coord))));
};

function pretty_coords($latitude, $longitude) { return sprintf("%s %s, %s %s", ($latitude>0)?"N":"S", pretty_coord($latitude), ($longitude>0)?"E":"W", pretty_coord($longitude)); };