Map projection

A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. 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.
 

Background

 
For simplicity of description, 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 surface having curvature in all three spatial dimensions.

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.

Carl Friedrich Gauss's Theorema Egregium proved that a sphere cannot be represented on a plane without distortion. Since any map projection is a represention of a sphere's surface 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.

A map of the Earth is a representation of a curved surface on a plane. Therefore a map projection must have been used to create the map, and, conversely, maps could not exist without map projections. 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.
 

Construction of a map projection

 
The creation of a map projection involves two steps:

1. Selection of a model for the shape of the Earth or planetary body (usually choosing between a sphere or ellipsoid). Because the Earth's actual shape is irregular, information is lost in this step.
2. Transformation of geographic coordinates (longitude and latitude) to Cartesian (x,y) or polar plane coordinates. Cartesian coordinates normally have a simple relation to eastings and northings defined on a grid superimposed on the projection.

Some of the simplest map projections are literally projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. This is not the case for most projections, which are defined only in terms of mathematical formulae that have no direct geometric interpretation.
 

Metric properties of maps

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

• Area
• Shape
• Direction
• Bearing
• Distance
• Scale

Map projections can be constructed to preserve one or more of these properties, though not all of them simultaneously. 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?

 
Due to the many uses of maps and the vast range of scales at which they are created, no single map projection serves well for all purposes. 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.

Thematic maps normally require an equal area projection so that phenomena per unit area are shown in correct proportion. 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.

While the mathematics of projection do not allow any particular map projection to stand out as an unqualified "best", on the other hand, the literature singles out the Mercator projection as having been overused and abused. The problem has long been recognized even outside professional circles: 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.

Likewise, the Peters map controversy motivated the American Cartographic Association (now Cartography and Geographic Information Society) to produce a series of booklets (including Which Map is Best) designed to educate the public about map projections and distortion in maps. In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution recommending against using any rectangular projection (including Mercator and Gall–Peters) for reference maps of the world.
 

Aspects of the projection

 
Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal (such that the surface's axis of symmetry coincides with the Earth's axis), transverse (at right angles to the Earth's axis) or oblique (any angle in between). The developable surface may also be either tangent or secant to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here.
 

Choosing a projection surface

 
A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface. The cylinder, cone and of course the plane are all developable surfaces. The sphere and ellipsoid are not developable surfaces. As noted in the introduction, any projection of a sphere or an ellipsoid onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing or warping it.)

One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion.
 

Scale

 
A globe is the only way to represent the earth with constant scale throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.

Some possible properties are:

• The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map.
• Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.
• Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the Mercator projection in normal aspect.
• Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the Azimuthal equidistant projection. There are also projections (Maurer, Close) where true distances from two points are preserved.

Choosing a model for the shape of the earth

 
Projection construction is also affected by how the shape of the Earth is approximated. In the following section on projection categories, the earth is taken as a sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.

Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict the land surface.

A third model of the shape of the Earth is the geoid, a complex and more accurate representation of the global mean sea level surface that is obtained through a combination of terrestrial and satellite gravity measurements. This model is not used for mapping because of its complexity, but rather is used for control purposes in the construction of geographic datums. (In geodesy, plural of "datum" is "datums" rather than "data".) A geoid is used to construct a datum by adding irregularities to the ellipsoid in order to better match the Earth's actual shape. It takes into account the large-scale features in the Earth's gravity field associated with mantle convection patterns, and the gravity signatures of very large geomorphic features such as mountain ranges, plateaus and plains.

Historically, datums have been based on ellipsoids that best represent the geoid within the region that the datum is intended to map. Controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic region (such as the North American Datum). A few modern datums, such as WGS84 which is used in the Global Positioning System, are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of accuracy in smaller regions.
 

Classification

 
A fundamental projection classification is based on the type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are cylindrical (e.g. Mercator), conic (e.g., Albers), or azimuthal or plane (e.g. stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.

Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are:

• Preserving direction (azimuthal), a trait possible only from one or two points to every other point
• Preserving shape locally (conformal or orthomorphic)
• Preserving area (equal-area or equiareal or equivalent or authalic)
• Preserving distance (equidistant), a trait possible only between one or two points and every other point
• Preserving shortest route, a trait preserved only by the gnomonic projection

Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.
 

Projections by surface

Cylindrical

 
The term "normal cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines.

The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation. This cylinder is wrapped around the Earth, projected onto, and then unrolled.

By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ):

• North-south stretching equals east-west stretching (secant φ): The east-west scale matches the north-south scale: conformal cylindrical or Mercator; this distorts areas excessively in high latitudes (see also transverse Mercator).
• North-south stretching grows with latitude faster than east-west stretching (secant² φ): The cylindric perspective (= central cylindrical) projection; unsuitable because distortion is even worse than in the Mercator projection.
• North-south stretching grows with latitude, but less quickly than the east-west stretching: such as the Miller cylindrical projection (secant[4φ/5]).
• North-south distances neither stretched nor compressed (1): equirectangular projection or "plate carrée".
• North-south compression precisely the reciprocal of east-west stretching (cosine φ): equal-area cylindrical. This projection has many named specializations differing only in the scaling constant. Some of those specializations are the Gall–Peters or Gall orthographic, Behrmann, and Lambert cylindrical equal-area). This kind of projection divides north-south distances by a factor equal to the secant of the latitude, preserving area at the expense of shapes.

In the first case (Mercator), the east-west scale always equals the north-south scale. In the second case (central cylindrical), the north-south scale exceeds the east-west scale everywhere away from the equator. Each remaining case has a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale.

Normal cylindrical projections map the whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width.
 

Pseudocylindrical

 
Pseudocylindrical projections represent the central meridian and each parallel as a single straight line segment, but not the other meridians. Each pseudocylindrical projection represents a point on the Earth along the straight line representing its parallel, at a distance which is a function of its difference in longitude from the central meridian.

• Sinusoidal: the north-south scale and the east-west scale are the same throughout the map, creating an equal-area map. On the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. Thus the shape of the map for the whole earth is the region between two symmetric rotated cosine curves.

The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map. The true distance between two points on the same parallel—and the true area of shapes on the map—are not distorted. The meridians drawn on the map help the user to realize the shape distortion and mentally compensate for it.

• Collignon projection, which in its most common forms represents each meridian as 2 straight line segments, one from each pole to the equator.
• Mollweide
• Goode homolosine
• Eckert IV
• Eckert VI 
• Kavrayskiy VII
• Tobler hyperelliptical

Hybrid

 
The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas.
 

Conical 

• Equidistant conic
• Lambert conformal conic
• Albers conic

Pseudoconical 

• Bonne
• Werner cordiform, upon which distances are correct from one pole, as well as along all parallels.
• Continuous American polyconic

Azimuthal (projections onto a plane)

 
Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. Usually these projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function r(d) of the true distance d, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map.

The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point.

The radial scale is r'(d) and the transverse scale r(d)/(R sin(d/R)) where R is the radius of the Earth.

Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane:

• The gnomonic projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan(d/R); a hemisphere already requires an infinite map,
• The General Perspective Projection can be constructed by using a point of perspective outside the earth. Photographs of Earth (such as those from the International Space Station) give this perspective.
• The orthographic projection maps each point on the earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; r(d) = c sin(d/R). Can display up to a hemisphere on a finite circle. Photographs of Earth from far enough away, such as the Moon, give this perspective.
• The azimuthal conformal projection, also known as the stereographic projection, can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan(d/2R); the scale is c/(2R cos²(d/2R)). Can display nearly the entire sphere on a finite circle. The full sphere requires an infinite map.

Other azimuthal projections are not true perspective projections:

• Azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the earth ( for the case where the tangent point is the North Pole, see the flag of the United Nations)
• Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the earth: r(d) = c sin(d/2R)
• Logarithmic azimuthal is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. r(d) = c ln(d/d₀); locations closer than at a distance equal to the constant d₀ are not shown

Projections by preservation of a metric property

Conformal

 
Conformal map projections preserve angles locally. These are some conformal projections:

• Mercator – rhumb lines are represented by straight segments
• Transverse Mercator
• Stereographic - shape of circles is conserved
• Roussilhe
• Lambert conformal conic
• Peirce quincuncial projection
• Adams hemisphere-in-a-square projection
• Guyou hemisphere-in-a-square projection

Equal-area

 
"Area preserving maps" redirects here. For the mathematical concept, see Measure-preserving dynamical system.

The equal-area Mollweide projection
 

 
These are some projections that preserve area:

• Gall orthographic (also known as Gall–Peters, or Peters, projection)
• Albers conic
• Lambert azimuthal equal-area
• Lambert cylindrical equal-area
• Mollweide
• Hammer
• Briesemeister
• Sinusoidal
• Werner
• Bonne
• Bottomley
• Goode's homolosine
• Hobo–Dyer
• Collignon
• Tobler hyperelliptical
• Snyder’s equal-area polyhedral projection, used for geodesic grids.

 

Equidistant

A two-point equidistant projection of Asia
 

 
These are some projections that preserve distance from some standard point or line:

• Equirectangular—distances along meridians are conserved
• Plate carrée—an Equirectangular projection centered at the equator
• Azimuthal equidistant—distances along great circles radiating from centre are conserved
• Equidistant conic
• Sinusoidal—distances along parallels are conserved
• Werner cordiform distances from the North Pole are correct as are the curved distance on parallels
• Soldner
• Two-point equidistant: two "control points" are arbitrarily chosen by the map maker. Distance from any point on the map to each control point is proportional to surface distance on the earth.

Gnomonic

 

The Gnomonic projection is thought to be the oldest map projection, developed by Thales in the 6th century BC
 

 
Great circles are displayed as straight lines:

• Gnomonic projection

Retroazimuthal

 
Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B:

• Littrow—the only conformal retroazimuthal projection
• Hammer retroazimuthal—also preserves distance from the central point
• Craig retroazimuthal aka Mecca or Qibla—also has vertical meridians

Compromise projections

 

The Robinson projection was adopted by National Geographic Magazine in 1988 but abandoned by them in about 1997 for the Winkel Tripel.

Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things "look right". Most of these types of projections distort shape in the polar regions more than at the equator. 

 

 
These are some compromise projections:

• Robinson
• van der Grinten
• Miller cylindrical
• Winkel Tripel
• Buckminster Fuller's Dymaxion
• B.J.S. Cahill's Butterfly Map
• Kavrayskiy VII
• Wagner VI
• Chamberlin trimetric
• Oronce Fine's cordiform

References

1. Choosing a World Map. Falls Church, Virginia: American Congress on Surveying and Mapping. 1988. p. 1. ISBN 0-9613459-2-6.
2. Slocum, Terry A.; Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard (2005). Thematic Cartography and Geographic Visualization (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. p. 166. ISBN 0-13-035123-7.
3. Bauer, H.A. (1942). "Globes, Maps, and Skyways (Air Education Series)". New York. p. 28
4. Miller, Osborn Maitland (1942). "Notes on Cylindrical World Map Projections". Geographical Review 43 (3): 405–409.
5. Raisz, Erwin Josephus. (1938). General Cartography. New York: McGraw–Hill. 2d ed., 1948. p. 87.
6. Robinson, Arthur Howard. (1960). Elements of Cartography, second edition. New York: John Wiley and Sons. p. 82.
7. Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections p. 157. Chicago and London: The University of Chicago Press. ISBN 0-226-76746-9. (Summary of the Peters controversy.)
8. American Cartographic Association's Committee on Map Projections, 1986. Which Map is Best p. 12. Falls Church: American Congress on Surveying and Mapping.
9. American Cartographer. 1989. 16(3): 222–223.
10. Snyder, John P. (1993). Flattening the earth: two thousand years of map projections. University of Chicago Press. ISBN 0-226-76746-9.
11. Snyder, John P. (1997). Flattening the earth: two thousand years of map projections. University of Chicago Press. ISBN 978-0-226-76747-5.
12. Weisstein, Eric W., "Sinusoidal Projection" from MathWorld.
13. Weisstein, Eric W., "Gnomonic Projection" from MathWorld.
14. "The Gnomonic Projection". Retrieved November 18, 2005.
15. Weisstein, Eric W., "Orthographic Projection" from MathWorld.
16. Weisstein, Eric W., "Stereographic Projection" from MathWorld.
17. Weisstein, Eric W., "Azimuthal Equidistant Projection" from MathWorld.
18. Weisstein, Eric W., "Lambert Azimuthal Equal-Area Projection" from MathWorld.
19. "http://www.gis.psu.edu/projection/chap6figs.html". Retrieved November 18, 2005.