Projected coordinate system

A projected coordinate system is a flat, two-dimensional representation of the Earth. It is based on a sphere or spheroid geographic coordinate system, but it uses linear units of measure for coordinates, so that calculations of distance and area are easily done in terms of those same units. The projected coordinate system is appropriate for regional data sets and applications.

The latitude and longitude coordinates are converted to x, y coordinates on the flat projection. The x coordinate is usually the eastward direction of a point, and the y coordinate is usually the northward direction of a point. The center line that runs east and west is referred to as the x axis, and the center line that runs north and south is referred to as the y axis.

The intersection of the x and y axes is the origin and usually has a coordinate of (0,0). The values above the x axis are positive, and the values below the x axis are negative. The lines parallel to the x axis are equidistant from each other. The values to the right of the y axis are positive, and the values to the left of the y axis are negative. The lines parallel to the y axis are equidistant.

Mathematical formulas are used to convert a three-dimensional geographic coordinate system to a two-dimensional flat projected coordinate system. The transformation is referred to as a map projection. Map projections usually are classified by the projection surface that is used, such as conic, cylindrical, and planar surfaces. Depending on the projection that is used, different spatial properties appear distorted. Projections are designed to minimize the distortion of one or two of the data's characteristics, yet the distance, area, shape, direction, or a combination of these properties might not be accurate representations of the data that is being modeled. There are several types of projections available. While most map projections attempt to preserve some accuracy of the spatial properties, others attempt to minimize overall distortion instead, such as the Robinson projection. The most common types of map projections include:

Equal area projections: These projections preserve the area of specific features. These projections distort shape, angle, and scale. The Albers Equal Area Conic projection is an example of an equal area projection.
Conformal projections: These projections preserve local shape for small areas. These projections preserve individual angles to describe spatial relationships by showing perpendicular graticule lines that intersect at 90 degree angles on the map. All of the angles are preserved; however, the area of the map is distorted. The Mercator and Lambert Conformal Conic projections are examples of conformal projections.
Equidistant projections: These projections preserve the distances between certain points by maintaining the scale of a specific data set. Some of the distances are true distances, which are the same distances at the same scale as the globe. If you go outside the data set, the scale becomes more distorted. The Sinusoidal projection and the Equidistant Conic projection are examples of equidistant projections.
True-direction or azimuthal projections: These projections preserve the direction from one point to all other points by maintaining some of the great circle arcs. These projections give the directions or azimuths of all points on the map correctly with respect to the center. Azimuthal maps can be combined with equal area, conformal, and equidistant projections. The Lambert Equal Area Azimuthal projection and the Azimuthal Equidistant projection are examples of azimuthal projections.