A map projection is a systematic mathematical method for representing the curved surface of the Earth, or another approximately spherical or ellipsoidal body, on a flat surface such as a printed map, screen, chart, or sheet of paper. The central problem is geometric. The Earth is three dimensional, whereas a conventional map is two dimensional, so a projection establishes a mathematical relationship between geographic coordinates on the Earth and positions on a plane.
In modern cartography, the source surface is normally an oblate spheroid or reference ellipsoid rather than a perfect sphere because the Earth is slightly flattened at the poles and has an irregular gravitational surface. A geographic position is commonly described using latitude and longitude, while a projected position is described using planar coordinates such as eastings and northings. The projection therefore provides transformation equations that convert one coordinate system into another. There is no single universally correct projection because flattening a curved surface necessarily changes at least some geometric properties. Every projection makes deliberate choices about which characteristics should be preserved most accurately and which distortions can be accepted.
The mathematical basis of a projection can be understood by imagining a coordinate grid covering the Earth and then transforming that grid onto a plane. In traditional descriptions, projections are often illustrated as projecting the Earth's surface onto a developable surface such as a cylinder, cone, or plane, although many modern projections are defined directly by equations and do not correspond literally to a physical projection apparatus.
Cylindrical projections commonly relate to a cylinder surrounding or intersecting the Earth, conic projections use a cone, and azimuthal projections use a plane touching or cutting the globe. Other projections, including the Robinson and Winkel Tripel projections, are compromise constructions whose equations are designed to balance several forms of distortion rather than to preserve one particular property perfectly. The projection may also be described according to its aspect, meaning the orientation of the projection relative to the Earth. A normal aspect may place a cylindrical or conic surface in a conventional orientation, while transverse and oblique aspects rotate the underlying geometry to suit particular mapping purposes.
The principal kinds of distortion produced by map projections concern area, shape, distance, and direction. An equal area projection preserves relative areas, so regions occupying twice the true area of another region continue to occupy twice the mapped area, although their shapes may become noticeably distorted. A conformal projection preserves local angles and therefore tends to preserve small shapes and bearings locally, but it cannot preserve area everywhere.
An equidistant projection preserves selected distances or distance relationships, usually from particular points or along particular lines rather than every distance on the map. An azimuthal projection can preserve directions from a specified central point, depending on its mathematical construction. A projection that does not preserve one of these properties exactly may still be highly useful. The terms conformal, equal area, equidistant, and azimuthal therefore describe mathematical properties rather than simple quality rankings. A projection can also have special lines or points where distortion is zero or particularly small, and distortion generally increases as locations move away from those areas.
Map projection terminology includes several related concepts that are important for interpreting maps correctly. A datum is a geodetic reference framework used to define positions on the Earth, whereas a projection specifies how those positions are transformed onto a plane. A coordinate reference system combines the necessary geodetic and coordinate information so that a location can be interpreted consistently. Geographic coordinates expressed as latitude and longitude are angular coordinates and are not themselves a map projection.
Projected coordinates normally use linear units such as metres or feet, depending on the coordinate reference system. A reference ellipsoid approximates the Earth's shape mathematically, while a geoid is a gravity related surface approximating mean sea level and is not simply interchangeable with an ellipsoid. Confusing a datum with a projection can produce positional errors because changing the underlying reference framework and changing the planar transformation are different operations. Modern geographic information systems therefore store projection and coordinate reference information explicitly so that datasets can be displayed, measured, transformed, and analysed correctly.
Map projections have a long history because mapmakers needed ways to represent the known world on navigable charts, regional maps, and world maps. Early cartographers developed geometric and mathematical methods for transforming spherical or approximately spherical geography onto planes, with important developments occurring in ancient Greek mathematics, medieval Islamic geography, Renaissance European cartography, navigation, surveying, and later geodesy. Gerardus Mercator's sixteenth century cylindrical conformal projection became especially important for navigation.
During the nineteenth and twentieth centuries, increasing mathematical understanding of the Earth's shape produced many specialised projections for national mapping, scientific analysis, aviation, oceanography, thematic cartography, and military applications. The development of computers made it practical to implement highly complex projection equations and to transform large datasets automatically. Modern digital cartography can reproject enormous quantities of geographic data, allowing the same underlying geographic information to be displayed in different coordinate systems according to the purpose of the map.
The choice of a projection depends primarily on what the map is intended to communicate. A navigation chart may prioritise local angles and bearings, making a conformal projection useful. A map comparing the sizes of countries, populations, ecosystems, agricultural regions, or other spatial quantities may require an equal area projection so that visual area remains meaningful. A regional map may use a projection specifically designed to minimise distortion over a particular continent, country, state, or survey region.
A world reference map may instead use a compromise projection that accepts moderate distortion in several categories to produce a visually balanced overall representation. Distance calculations require particular care because measuring a straight line on a projected map does not automatically give the true distance on the Earth's surface. Geographic information systems can calculate geodesic distances or use suitable projected coordinate systems when accurate planar measurement is required. Consequently, a projection should be selected according to the spatial extent, geometry, analytical purpose, and properties that matter most.
A common misconception is that a projection merely changes the visual appearance of a map while leaving every geographic relationship unchanged. In reality, the transformation can substantially affect apparent size, shape, distance, direction, spacing, and visual emphasis. The familiar appearance of a world map can therefore influence people's intuitive understanding of geography.
For example, projections that enlarge high latitude regions can make Greenland appear much larger relative to Africa than their true areas justify, while equal area projections may produce unfamiliar shapes that nevertheless represent areas more faithfully. Another misconception is that distortion means a map is inaccurate in every respect. A projection may be deliberately and mathematically accurate for the property it was designed to preserve while sacrificing other properties. The appropriate question is therefore not whether a projection has distortion, because all flat world maps necessarily do, but what kind of distortion it has, where that distortion occurs, how large it is, and whether it matters for the intended use.
In mathematical terms, a map projection is a transformation from coordinates on a curved reference surface to coordinates in a plane. If geographic coordinates are represented by latitude and longitude, a projection can be represented abstractly by functions that transform those angular coordinates into planar coordinates, although the exact equations vary substantially between projections. The transformation can be examined through scale factors, angular deformation, area ratios, and other measures of distortion.
Tissot's indicatrix is a classical cartographic method for visualising local distortion by showing how an infinitesimally small circle on the Earth's surface becomes an ellipse or another altered figure on a projected map. The orientation and dimensions of that ellipse reveal information about local scale and directional distortion. Projection parameters can include a central meridian, latitude of origin, standard parallels, central scale factor, false easting, and false northing. These parameters allow a general projection family to be adapted to a particular region or coordinate reference system. Map projection is therefore both a cartographic concept and a rigorous mathematical component of geographic information science.
