#jsDisabledContent { display:none; } My Account |  Register |  Help

# Map projection

Article Id: WHEBN0000051784
Reproduction Date:

 Title: Map projection Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Map projection

A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy's Geography and using his second map projection

Commonly a map projection is a systematic transformation of the latitudes and longitudes of locations on the surface of a sphere or an ellipsoid into locations on a plane.[1] Map projections are necessary for creating maps. All map projections distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.

More generally, the surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid; see below. Even more generally, projections are the subject of several pure mathematical fields, including differential geometry and projective geometry. However "map projection" refers specifically to a cartographic projection.

## Contents

• Background 1
• Metric properties of maps 2
• Which projection is best? 2.1
• Distortion 2.2
• Construction of a map projection 3
• Choosing a projection surface 3.1
• Aspects of the projection 3.2
• Scale 3.3
• Choosing a model for the shape of the body 3.4
• Classification 4
• Projections by surface 5
• Cylindrical 5.1
• Pseudocylindrical 5.2
• Hybrid 5.3
• Conic 5.4
• Pseudoconic 5.5
• Azimuthal (projections onto a plane) 5.6
• Projections by preservation of a metric property 6
• Conformal 6.1
• Equal-area 6.2
• Equidistant 6.3
• Gnomonic 6.4
• Retroazimuthal 6.5
• Compromise projections 6.6
• References 8

## Background

Maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the Earth's surface at once; and they are cheaper to produce and transport. These useful traits of maps motivate the development of map projections.

However, Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort. Every distinct map projection distorts in a distinct way. The study of map projections is the characterization of these distortions.

Projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection. Few projections in actual use are perspective.

For simplicity most of this article assumes that the surface to be mapped is that of a sphere. In reality, the Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes. These other surfaces can be mapped as well. Therefore, more generally, a map projection is any method of "flattening" into a plane a continuous curved surface.

## Metric properties of maps

An Albers projection shows areas accurately, but distorts shapes.

Many properties can be measured on the Earth's surface independently of its geography. Some of these properties are:

Map projections can be constructed to preserve at least one of these properties, though only in a limited way for most. Each projection preserves or compromises or approximates basic metric properties in different ways. The purpose of the map determines which projection should form the base for the map. Because many purposes exist for maps, many projections have been created to suit those purposes.

Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information; their collection depends on the chosen datum (model) of the Earth. Different datums assign slightly different coordinates to the same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection. The slight differences in coordinate assignation between different datums is not a concern for world maps or other vast territories, where such differences get shrunk to imperceptibility.

### Which projection is best?

The mathematics of projection do not permit any particular map projection to be "best" for everything. Something will always get distorted. Therefore a diversity of projections exists to service the many uses of maps and their vast range of scales.

Modern national mapping systems typically employ a transverse Mercator or close variant for large-scale maps in order to preserve conformality and low variation in scale over small areas. For smaller-scale maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose.[2]

Thematic maps normally require an equal area projection so that phenomena per unit area are shown in correct proportion.[3] However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area. Hence reference maps of the world often appear on compromise projections instead. Due to the severe distortions inherent in any map of the world, within reason the choice of projection becomes largely one of æsthetics.

The Mercator projection, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate.[4][5][6][7] This problem has long been recognized even outside professional circles. For example a 1943 New York Times editorial states:
The time has come to discard [the Mercator] for something that represents the continents and directions less deceptively... Although its usage... has diminished... it is still highly popular as a wall map apparently in part because, as a rectangular map, it fills a rectangular wall space with more map, and clearly because its familiarity breeds more popularity.[8]

A controversy in the 1980s over the

• A Cornucopia of Map Projections, a visualization of distortion on a vast array of map projections in a single image.
• G.Projector, free software can render many projections (NASA GISS).
• Color images of map projections and distortion (Mapthematics.com).
• Geometric aspects of mapping: map projection (KartoWeb.itc.nl).
• Java world map projections, Henry Bottomley (SE16.info).
• Map projections http://www.3dsoftware.com/Cartography/USGS/MapProjections/ at the Wayback Machine (archived January 4, 2007) (3DSoftware).
• Map projections, John Savard.
• Map Projections (MathWorld).
• Map Projections An interactive JAVA applet to study deformations (area, distance and angle) of map projections (UFF.br).
• Map Projections: How Projections Work (Progonos.com).
• Map Projections Poster (U.S. Geographical Survey).
• MapRef: The Internet Collection of MapProjections and Reference Systems in Europe
• PROJ.4 – Cartographic Projections Library.
• Projection Reference Table of examples and properties of all common projections (RadicalCartography.net).
• Understanding Map Projections PDF (1.70 MB), Melita Kennedy (ESRI).
• World Map Projections, Stephen Wolfram based on work by Yu-Sung Chang (Wolfram Demonstrations Project).