Angular Units Thumbnail

A couple of weeks ago, Aileen wrote a blog post called "About geographic transformations and how to choose the right one". In it, she described many of the parameters that you can set for map projections.  Two that were not mentioned were angular and linear units, so I thought it might help to describe them here.

Angular units in the projection file

In most Geographic Coordinate System projection files, such as GCS_North_American_1927.prj (shown below), you have a unit of measure described as a "Degree", with the value 0.017453292519943295.

GEOGCS["GCS_North_American_1927",DATUM["D_North_American_1927",SPHEROID ["Clarke_1866",6378206.4,294.9786982]],PRIMEM["Greenwich",0],UNIT ["Degree",0.017453292519943295]]

Trigonometric functions are used in geodesy to calculate coordinates on a datum and spheroid in order to perform geographic transformations between datums and other operations. Standard trigonometric functions, however, do not use units of degrees – these calculations are performed with units of radians.

The circumference of a circle, from basic geometry, is defined as pi times twice the radius of the circle, or C = 2pi r. However, this yields a circumference in linear units, like feet or meters. The trig functions used in operations on a spheroid and datum can only use angular units of radians so these trig functions will not work.

Instead of thinking of the circle in terms of a circumference measured in linear units like feet or meters, ArcGIS internally converts the circumference to radians, a different angular unit.

The circumference of a circle in radians equals 2pi.

The value in the projection file for a Degree is the number of radians in a degree. Here is the calculation:

2pi (radians in a circle) / 360° = 0.017453292519943295769236907684886 radians per degree

Since ArcGIS Desktop calculates coordinate position to 16 significant digits, the value of 0.017453292519943295, rounded to 18 places to the right of the decimal, is sufficient to maintain data accuracy when performing these calculations.

Having converted the angular units of measure from degrees to radians, internal trigonometric calculations can be performed using standard formulas.

(Thanks to Rob Juergens of the ArcGIS Projection Engine Team for explaining this – I’ve wondered about this value for "degree" for several years.)

Linear units in the projection file

A projected coordinate system expresses the location of data using linear units that can be measured on the ground with a ruler. Projected coordinate systems are used when distance or area must be calculated for data. The most commonly used linear units are FEET or METERS, although other linear units can also be used. In the United States, two different definitions of the foot are typically used:

The U.S. Survey Foot, describes as the unit foot_us in ArcGIS Desktop. This unit is equal to "exactly" 1200/3937 of a meter, or 0.30480060960121920243840487680975...

The International Foot is also used in ArcGIS Desktop, expressed as the unit foot. Some states in the US have said that this unit can also be used with the State Plane Coordinate System, since this definition of a foot will convert exactly to a meter. The International Foot is equal to exactly 0.3048 of a meter.

The unit meter has been redefined many times over the years. The standard definition of the meter today is based on the speed of light in a vacuum: exactly 299,792,458 meters per second. The meter is defined as the distance light travels in a vacuum in exactly 1/299,792,458 of a second, and is equal to 3.2808333333333333333333333333333 U.S. Survey Feet (3937/1200) or 39.37 U. S. Survey Inches.