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While all geospatial data needs to be georeferenced (pinned to locations on the Earth's surface) in some way, a given data set might or might not explicitly describe locations with geographic coordinates (latitudes and longitudes). When it does, many applications—particularly map display—cannot make direct use of geographic coordinates, and must transform them in some way to plane coordinates. This transformation process, called map projection, is both algorithmic and the core of the cartographer's art.
A map projection is a procedure that unwraps a sphere or ellipsoid to flatten it onto a plane. Usually this is done through an intermediate surface such as a cylinder or a cone, which is then unwrapped to lie flat. Consequently, map projections are classified as cylindrical, conical, and azimuthal (a direct transformation of the surface of part of a spheroid to a circle). See The Three Main Families of Map Projections for discussions and illustrations of how these transformations work.
Mapping Toolbox™ map projection libraries feature dozens of map projections, which you principally control with axesm. Some are ancient and well-known (such as Mercator), others are ancient and obscure (such as Bonne), while some are modern inventions (such as Robinson). Some are suitable for showing the entire world, others for half of it, and some are only useful over small areas. When geospatial data has geographic coordinates, any projection can be applied, although some are not good choices. The toolbox can project both vector data and raster data.
For more information on projection classes, functions and definitions see Projections and Parameters
When geospatial data has plane coordinates (i.e., it comes preprojected, as do many satellite images and municipal map data sets), it is usually possible to recover geographic coordinates if the projection parameters and datum are known. Using this information, you can perform an inverse projection, running the projection backward to solve for latitude and longitude. The toolbox can perform accurate inverse projections for any of its projection functions as long as the original projection parameters and reference ellipsoid (or spherical radius) are provided to it.
Note: Converting a position given in latitude-longitude to its equivalent in a projected map coordinate system involves converting from units of angle to units of length. Likewise, unprojecting a point position changes its units from those of length to those of angle). Unit conversion functions such as deg2km and km2deg also convert coordinates between angles and lengths, but do not transform the space they inhabit. You cannot use them to project or unproject coordinate data.
All map projections introduce distortions compared to maps on globes. Distortions are inherent in flattening the sphere, and can take several forms:
Areas — Relative size of objects (such as continents)
Distances — Relative separations of points (such as a set of cities)
Directions — Azimuths (angles between points and the poles)
Shapes — Relative lengths and angles of intersection
Some classes of map projections maintain areas, and others preserve local shapes, distances, and/or directions. No projection, however, can preserve all these characteristics. Choosing a projection thus always requires compromising accuracy in some way, and that is one reason why so many different map projections have been developed. For any given projection, however, the smaller the area being mapped, the less distortion it introduces if properly centered. Mapping Toolbox tools help you to quantify and visualize projection distortions.