Map projection
A map projection is a geometric function that transforms the earth's curved, ellipsoidal surface onto a flat, 2dimensional plane.^{[1]} Since the Earth is roughly the shape of an oblate spheroid, map projections are necessary for creating maps of the Earth or parts of the Earth that are represented on a plane such as a piece of paper or a computer screen.
A map projection is an essential component of any modern map, and there are an infinite number of possible map projections. Since Gerardus Mercator presented his Mercator global map projection in 1569, numerous map projections have been developed and scores of projections are currently used by cartographers today.
Contents
Background
Carl Friedrich Gauss's Theorema Egregium ^{[2]} proved that a sphere cannot be represented on a plane without distortion. Since any method of representing a sphere's surface on a plane is a map projection, 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 motivated the development of map projections.
Metric properties of maps
Map projections can be used in a map to preserve one or more of the following properties, though never all of them simultaneously:
Each projection preserves, compromises, or approximates these properties in different ways. Some map projections are clearly better for specific purposes than others, so the purpose of a map must be considered to determine which projection should be used. To select a map projection, determine which of the properties is the most important for the project and select the coordinate system or map projection that best preserves that property. This decision normally involves allowing for some type of distortion to occur in order to minimize or eliminate distortion for one or more other properties that are essential to the map's objective.
Another major concern that drives the choice of a projection is the compatibility of data sets (geographic information). As such, their collection depends on the chosen model of the Earth. Different models assign slightly different coordinates to the same location, so it is important that the model be known and that the chosen projection be compatible with that model. On small areas (large scale), data compatibility issues are less important since metric distortions are minimal at this level. In very large areas (small scale), distortion is a more important factor to consider.
Choosing a Proper Projection
Choosing the proper projection for a map is vital to correctly presenting the map's information. A proper projection helps the viewer recognize the map's area of best representation; the misuse of projection (either intentionally or unintentionally) gives viewers a skewed perspective of the area they are viewing. A number of principles govern choosing the proper projection. ^{[3]}
 Scale: Projections are more important for smallscale maps (those that cover large areas); world maps are particularly important.
 Size and Shape of the Area: different projections should be chosen for areas on a NorthSouth versus EastWest orientation.
 Location: latitudinal location of the map area determines what type of projection should be used (example, a conic for midlatitudes).
 Audience: the sophistication of the map user should be considered when determining the complexity of the projection.
Classification of Projections
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 (meridians are arcs of circles), pseudocylindrical (meridians are straight lines), 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 (equalarea 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.
NOTE: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equalarea and conformal.
There are also times when working with large scales, such as districts or provinces within countries, that distortion doesn't play a significant role, in which any projection that is centered on the area of interest is acceptable.
Projections by surface
Cylindrical projection
The term "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 (or, mutatis mutandis, more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines).
The mapping of meridians to vertical lines can be visualized by imagining a cylinder (of which the axis coincides with the Earth's axis of rotation) wrapped around the Earth and then projecting onto the cylinder, and subsequently unfolding the cylinder.
By the geometry of their construction, cylindrical projections stretch distances eastwest. The amount of stretch 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 northsouth stretching (where latitude is given by φ):
 Northsouth stretching is equal to the eastwest stretching (secant φ): The eastwest scale matches the northsouth scale: conformal cylindrical or Mercator; this distorts areas excessively in high latitudes (see also transverse Mercator).
 Northsouth stretching growing rapidly with latitude, even faster than eastwest stretching (secant² φ: The cylindric perspective (= central cylindrical) projection; unsuitable because distortion is even worse than in the Mercator projection.
 Northsouth stretching grows with latitude, but less quickly than the eastwest stretching: such as the Miller cylindrical projection (secant[4φ/5]).
 Northsouth distances neither stretched nor compressed (1): equidistant cylindrical or plate carrée.
 Northsouth compression precisely the reciprocal of eastwest stretching (cosine φ): equalarea cylindrical (with many named specializations such as GallPeters projection or Gall orthographic, Behrmann, and Lambert cylindrical equalarea). This divides northsouth distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes.
In the first case (Mercator), the eastwest scale always equals the northsouth scale. In the second case (central cylindrical), the northsouth scale exceeds the eastwest 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 eastwest scale matches the northsouthscale.
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. This causes a distortion in the northern and southern parts of the map, making it appear as if the area of the land in those regions is larger than it actually is. For this reason, while cylindrical map projections are useful in mapping comparisons at similar latitudes, they should not be used for comparing things at different latitudes as they would not be projected on the same scale and thus provide inaccurate information.
Cylindrical projections are commonly used for world maps, regions bordering the equator, and regions that are predominantly northsouth in extent.^{[5]}
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. Because the meridians are not always perpendicular to the parallels, conformality is lost, leading to extreme distortion in shape at the poles. Area preservation is the goal using this map projection.
 Sinusoidal: the northsouth scale and the eastwest scale are the same throughout the map, creating an equalarea 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.^{[6]}
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.
Other Projections:^{[7]}
Secant Projections
Most map projections can become a secant projection. A good way to describe a secant projection is to see what happens when a cylindrical projection becomes a secant projection. A typical cylindrical projection has one line of tangency around the earth. If the cylinder circumference is smaller, the cylinder would pass through the earth and out the other side. This creates two secant, or standard, lines on the reference globe. There is no distortion at the secant lines. Distortion has less of an impact on certain areas on the map, specifically on and around the secant lines. However, distortion is increased in other areas. For example, in a Euler conic projection, secant lines in the northern hemisphere decrease area distortion in the northern hemisphere, while the higher latitudes in the southern hemisphere are more distorted. Choosing to use either secant lines or a tangent line depends on if there needs to be an area with limited distortion on the map. It should be noted that cylindrical projections are not the only projections that can be either tangential or secantial.^{[8]} ^{[9]}
Conical
Conic projections are geometrically based on projecting the earth's surface on a cone that has been wrapped around the globe. When produced using the normal (and most common) polar aspect, meaning the tip of the cone lies directly over one of the earth’s poles, conic projections feature two main characteristics: meridians that converge at a pole and come together at a regular rate, and parallels that cross meridians at right angles with map distortion equal along the parallels. Conic projections have at least one Standard parallel (where the map is tangent to the earth’s surface), but often have two surrounding the area being mapped to reduce distortion (map is secant to the earth’s surface). Due to their distortion pattern and location of standard parallels, conic projections are often used to map temperate areas of the world such as the United States, Europe, or Russia. ^{[10]} ^{[11]} Conical projections are best used to map relatively small areas because a small area map limits conical projection distortion.Types of Conical Projections include:
 Equidistant Conic projection
 Lambert conformal conic
 Lambert azimuthal equalarea projection
 Albers conic
 Polyconic Projections
Pseudoconical projections
Pseudoconic projections are similar to conic projections in a way that their parallels are partial concentric circles. The difference from conic projection is that meridians are curved rather than straight. The meridians are equally spaced. The outline is approximately heartshaped.^{[12]}
 Bonne in this projection the parallels of latitude are concentric circular arcs, and the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted. The Bonne method has fallen into disuse, giving way to the transverse Mercator method. However, it is still a credible way to preserve shapes and areas. Distortions are are noticeable the further away from the center of the map. The scale is only accurate along the vertical axis.
 Werner cordiform designates a pole and a meridian; distances from the pole are preserved, as are distances from the meridian (which is straight) along the parallels
 Continuous American polyconic refers to those projections whose parallels are all nonconcentric circular arcs, except for a straight equator, and the centers of these circles lie along a central axis. This description applies to projections in equatorial aspect.^{[13]}
Azimuthal projections
Azimuthal projections have the property that directions from a central point are preserved (and hence, 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.^{[14]}^{[15]}
 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).^{[16]} 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)).^{[17]} 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^{[18]} (for the case where the tangent point is the North Pole, see the flag of the United Nations)
 Lambert azimuthal equalarea. Distance from the tangent point on the map is proportional to straightline distance through the earth: r(d) = c sin(d/2R)^{[19]}
 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. Works well with cognitive maps. r(d) = c ln(d/d_{0}); locations closer than at a distance equal to the constant d_{0} are not shown.^{[20]}
Azimuthal projections are commonly used to map polar regions, roughly circular regions, or for maps of continents and hemispheres. ^{[21]}
Projections by preservation of a metric property
Conformal
Conformal, or orthomorphic, projections are those on which any small area has the same shape as on the globe: rectangles remain rectangles, and the relation between the length of parallels and meridians is the same as on the globe. That is accomplished by keeping the angle between any two intersecting lines the same on both the flat map and the globe. A conformal projection can significantly distort the shape of long geographic features, however within a small area of the point of intersection, the shape will be correct and the scale will be the same in all direction. Although all map projections distort the shapes of large territories like continents, in general a conformal projection offers a less distorted picture of gross shape than a nonconformal projection.
Unfortunately, however, in order to construct such a projection, the scale has to be varied considerably.^{[22]}
Examples of projections that are conformal include:
 Mercator  rhumb lines are represented by straight segments
 Stereographic  shape of circles is conserved
 Roussilhe
 Lambert conformal conic
 Quincuncial map
 Guyou hemisphereinasquare projection
Equalarea
A map designed around an equalarea (equivalent) projection preserves the property of area so that any area measured on the map is the same as it is measured on the Earth. Equalarea projections are wellsuited for maps of general interest and for those showing distributions over space, such as population, wildlife habitats, and land cover.^{[23]} An equalarea projection, however, causes distortions in shape and distance. These projections preserve area:
 Gall orthographic (also known as GallPeters, or Peters, projection)
 Albers equalarea conic
 Lambert azimuthal equalarea
 Mollweide projection
 Hammer
 Briesemeister
 Sinusoidal
 Werner
 Bonne
 Bottomley
 Goode homolosine
 HoboDyer
 Collignon
 Tobler hyperelliptical
Equidistant
An equidistant map shows distances correctly. However, this is not attainable over an entire map; it is only correct along certain lines or from a specific point. These lines of true scale are called central meridians for cylindrical projections, and standard parallels for conic projections.^{[24]} The followling are some examples of projections that attempt to keep distances correct:
 Equidistant Cylindrical
 Equirectangular projection  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 distances along the meridians are conserved and commonly along 2 standard parallels
 Sinusoidal  distances along parallels are conserved
 Werner cordiform distances from the North Pole are correct as are the curved distance on parallels
 Soldner
 Twopoint 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
Projected using the center of the Earth as an origin. Great circles are displayed as straight lines:
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
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:
 Robinson projection
 van der Grinten
 Miller Cylindrical
 Winkel Tripel
 Fuller Projection or Dymaxion Map
 Butterfly World Map
 Steve Waterman's Butterfly Map
 Kavrayskiy VII
 Wagner VI
Types of Map Projection
The Earth is spherical in nature and to accurately represent the shape, area, distance, and direction (SADD) of the entire or part of the Earth's surface in a single map projection is not possible. The following table describes the various map projections and their uses.
List of map projections used to represent the Earth's surface
Type of projection  Brief description 
Aitoff  A compromise projection developed in 1889 and used for world maps.^{[25]} 
Alaska Grid  This projection was developed to provide a conformal map of Alaska with less scale distortion than other conformal projections.^{[25]} 
Alaska Series E  Developed in 1972 by the United States Geological Survey (USGS) to publish a map of Alaska at 1:2,500,000 scale.^{[25]} 
Albers Equal Area Conic  This conic projection uses two standard parallels to reduce some of the distortion of a projection with one standard parallel. Shape and linear scale distortion are minimized between the standard parallels.^{[25]} 
Armadillo projection  The Armadillo projection (also called the Raisz Armadillo projection or Orthoapsidal projection) is a world map projection developed by Erwin Raisz in 1943 that projects orthographically onto the surface of a torus with radii 1 and 1. 
Authalic Projection  An authalic map projection is one in which at any point the scales in two orthogonal directions are inversely proportional.^{[26]} 
Azimuthal Equidistant  The most significant characteristic of this projection is that both distance and direction are accurate from the central point.^{[25]} 
Babinet  Also called 'homalographic', Mollweide, or Babinet projection. A world map projection that minimizes area distortion while sacrificing the accuracy of shape and angle. It is a representation of the world map on an ellipse with a 2:1 ratio. 
Berghaus Star  The Berghaus Star projection is a starshaped azimuthal world map projection that is neither conformal nor equal area, however the Northern Hemisphere of the Earth in this projection is azimuthal equidistant. 
Bipolar Oblique Conformal Conic  This projection was developed specifically for mapping North and South America and maintains conformality.^{[25]} 
Bonne  This equalarea projection has true scale along the central meridian and all parallels.^{[25]} 
Cassini–Soldner  This transverse cylindrical projection maintains scale along the central meridian and all lines parallel to it. This projection is neither equal area nor conformal. It is most suited for large scale mapping of areas predominantly northsouth in extent.^{[27]} 
Chamberlin Trimetric  This projection was developed and used by the National Geographic Society for continental mapping. The distance from three input points to any other point is approximately correct.^{[25]} 
Craster Parabolic  This pseudocylindrical equalarea projection is primarily used for thematic maps of the world.^{[25]} 
Cylindrical Equal Area  See Equal Area Cylindrical 
Double Stereographic  This azimuthal projection is conformal.^{[25]} 
Eckert I  This pseudocylindrical projection is used primarily as a novelty map.^{[25]} 
Eckert II  A pseudocylindrical equalarea projection.^{[25]} 
Eckert III  This pseudocylindrical projection is used primarily for world maps.^{[25]} 
Eckert IV  This pseudocylindrical projection projection is used primarily for world maps.^{[25]} 
Eckert V  This pseudocylindrical projection is used primarily for world maps.^{[25]} 
Eckert VI  This equalarea projection is used primarily for world maps.^{[25]} 
Equidistant Conic  This conic projection can be based on one or two standard parallels. As the name implies, all circular parallels are spaced evenly along the meridians.^{[25]} 
Equidistant Cylindrical  One of the easiest projections to construct because it forms a grid of equal rectangles.^{[25]} 
Equal Area Cylindrical  This projection is an equalarea cylindrical projection suitable for world mapping.^{[25]} 
Equirectangular  This projection is very simple to construct because it forms a grid of equal rectangles.^{[25]} 
Fuller  A projection of a global map onto the surface of a polyhedron, which, when expanded to a flat, twodimensional map, retains most of the relative proportional integrity (relative size and shape) of global features. 
Gall Stereographic  The Gall Stereographic cylindrical projection results from projecting the earth's surface from the equator onto a secant cylinder intersected by the globe at 45° N and 45° S. This projection moderately distorts distance, shape, direction, and area.^{[28]} 
Gauss–Krüger  This projection is similar to the Mercator except that the cylinder is tangent along a meridian instead of the equator. The result is a conformal projection that does not maintain true directions.^{[25]} 
Geocentric Coordinate System  The geocentric coordinate system is not a map projection. The earth is modeled as a sphere or spheroid in a righthanded X,Y,Z system.^{[25]} 
Geographic Coordinate System  The geographic coordinate system is not a map projection. The earth is modeled as a sphere or spheroid.^{[25]} 
Gnomonic  This azimuthal projection uses the center of the earth as its perspective point.^{[25]} 
Goode homolosine  A pseudocylindrical, equalarea map projection used for world maps. Its equalarea makes it useful for raster data representation. 
Great Britain National Grid  This coordinate system uses a Transverse Mercator projected on the Airy spheroid. The central meridian is scaled to 0.9996. The origin is 49° N and 2° W.^{[25]} 
Hammer–Aitoff  The HammerAitoff equalarea projection, also called the Hammer projection, is a map projection that is a modification of the Lambert azimuthal equalarea projection. It consists of halving the vertical coordinates of the equatorial aspect of one hemisphere and doubling the values of the meridians from the center (Snyder 1987, p. 182). Like the Lambert azimuthal equalarea projection, it is equal area, but it is no longer azimuthal. It is best used for countries that have a long axis, but not an extreme long axis.^{[29]}^{[30]} 
Hotine Oblique Mercator  This is an oblique rotation of the Mercator projection. Developed for conformal mapping of areas that do not follow a north–south or east–west orientation but are obliquely oriented.^{[25]} 
Krovak  The Krovak projection is an oblique Lambert conformal conic projection designed for the former Czechoslovakia.^{[25]} 
Lambert Azimuthal Equal Area  This projection preserves the area of individual polygons while simultaneously maintaining true directions from the center.^{[27]} 
Lambert Conformal Conic  This projection is one of the best for middle latitudes. It is similar to the Albers Conic Equal Area projection except that the Lambert Conformal Conic projection portrays shape more accurately than area.^{[25]} 
Local Cartesian Projection  This is a specialized map projection that does not take into account the curvature of the earth.^{[25]} 
Loximuthal  This projection shows loxodromes, or rhumb lines, as straight lines with the correct azimuth and scale from the intersection of the central meridian and the central parallel.^{[25]} 
McBryde–Thomas FlatPolar Quartic  This equalarea projection is primarily used for world maps.^{[25]} 
Mercator  Originally created to display accurate compass bearings for sea travel. An additional feature of this projection is that all local shapes are accurate and clearly defined.^{[27]} 
Miller Cylindrical  This projection is similar to the Mercator projection except that the polar regions are not as distorted.^{[27]} 
Mollweide  Pseudocylindrical and equalarea; created for world maps. The central meridian is straight and the 90th meridians are circular arcs. Parallels are straight, but unequally spaced. Scale is true only along the standard parallels of 40:44 N and 40:44 S.^{[28]} 
New Zealand National Grid  This is the standard projection for largescale maps of New Zealand.^{[25]} 
Oblique Aspect Orthographic  Provides perspective views of hemispheres. Area and shape are distorted. Distances are true along the equator and other parallels.^{[28]} 
Oblique Mercator  Oblique Mercator projections are used to portray regions along great circles. Distances are true along a great circle defined by the tangent line formed by the sphere and the oblique cylinder, elsewhere distance, shape, and areas are distorted.^{[28]} 
Orthographic  This perspective projection views the globe from an infinite distance. This gives the illusion of a threedimensional globe.^{[25]} 
Orthoapsidal  See Armadillo projection 
Perspective  This projection is similar to the Orthographic projection in that its perspective is from space. In this projection, the perspective point is not an infinite distance away; instead, you can specify the distance.^{[25]} 
Peters (or GallPeters)  A modified equalarea cylindrical projection presented by German political propagandist Arno Peters as original, but is actually identical to the Gall projection.^{[28]} 
Plate Carrée  This projection is very simple to construct because it forms a grid of equal rectangles.^{[25]} 
Polar Stereographic  The projection is equivalent to the polar aspect of the Stereographic projection on a spheroid. The central point is either the North Pole or the South Pole.^{[25]} 
Polyconic  The name of this projection translates into ‘many cones’ and refers to the projection methodology; used for most of the earlier USGS topographic quadrangles. Based on an infinite number of cones tangent to an infinite number of parallels. The central meridian is straight. Other meridians are complex curves. The parallels are nonconcentric circles. Scale is true along each parallel and along the central meridian.^{[28]} 
Quartic Authalic  This pseudocylindrical equalarea projection is primarily used for thematic maps of the world.^{[25]} 
Rectified Skewed Orthomorphic  This oblique cylindrical projection is provided with two options for the national coordinate systems of Malaysia and Brunei.^{[25]} 
Robinson  A pseudocylindrical map projection which distorts shape, area, scale and distance to create attractive average projection properties.^{[29]} 
Secant Projection  A secant projection contains two secant lines to increase the area of best representation. ^{[31]} Albers Equal Area Conic projection and the Lambert Conformal Conic projection are both secant projections. 
Simple Conic  This conic projection can be based on one or two standard parallels.^{[25]} 
Sinusoidal  As a world map, this projection maintains equal area despite conformal distortion.Also known as the 'MercatorSansonFlamsteed' projection.^{[25]} 
Space Oblique Mercator  This projection is nearly conformal and has little scale distortion within the sensing range of an orbiting mapping satellite such as Landsat.^{[25]} 
State Plane Coordinate System (SPCS)  The State Plane Coordinate System is not a projection. It is a coordinate system that divides the 50 states of the United States, Puerto Rico, and the U.S. Virgin Islands into more than 120 numbered sections, referred to as zones.^{[25]} 
Stereographic  This azimuthal projection is conformal.^{[25]} 
Times  The Times projection was developed by Moir in 1965 for Bartholomew Ltd., a British mapmaking company. It is a modified Gall’s Stereographic, but the Times has curved meridians.^{[25]} 
Transverse Mercator  Similar to the Mercator except that the cylinder is tangent along a meridian instead of the equator. The result is a conformal projection that does not maintain true directions.^{[25]} 
TwoPoint Equidistant  This modified planar projection shows the true distance from either of two chosen points to any other point on a map.^{[25]} 
Universal Polar Stereographic (UPS)  This form of the Polar Stereographic maps areas north of 84° N and south of 80° S that are not included in the UTM Coordinate System. The projection is equivalent to the polar aspect of the Stereographic projection of the spheroid with specific parameters.^{[25]} 
Universal Transverse Mercator (UTM)  The Universal Transverse Mercator coordinate system is a specialized application of the Transverse Mercator projection. The globe is divided into 60 zones, each spanning six degrees of longitude.^{[25]} 
Van Der Grinten I  This projection is similar to the Mercator projection except that it portrays the world as a circle with a curved graticule.^{[25]} 
Vertical NearSide Perspective  Unlike the Orthographic projection, this perspective projection views the globe from a finite distance. This perspective gives the overall effect of the view from a satellite.^{[25]} 
Winkel I  A pseudocylindrical projection used for world maps that averages the coordinates from the Equirectangular (Equidistant Cylindrical) and Sinusoidal projections.^{[25]} 
Winkel II  A pseudocylindrical projection that averages the coordinates from the Equirectangular and Mollweide projections.^{[25]} 
Winkel Tripel  A compromise projection used for world maps that averages the coordinates from the Equirectangular (Equidistant Cylindrical) and Aitoff projections.^{[25]} 
Construction of a map projection
The creation of a map projection involves three steps:
 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.
 Transformation of geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y)
 Reduction of the scale (it does not matter in what order the second and third steps are performed)
Most map projections are not "projections" in any physical sense. Rather, they depend on mathematical formulae that have no direct physical interpretation. However, in understanding the concept of a map projection it can be helpful to think of a globe with a light source placed at some definite point relative to it, projecting features of the globe onto a surface. The following discussion of developable surfaces is based on that concept.
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, you 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.
Orientation of the projection
Once a choice is made between projecting onto a cylinder, cone, or plane, the orientation of the shape must be chosen. The orientation is how the shape is placed relative to the globe. The orientation of the projection surface can 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).
Case and tangency
Case refers to the map projection’s interaction with the portion of the earth’s surface depicted on the map. Each map has a point, line, or lines of tangency. In cartography, tangency refers to the point, line, or lines where the globe touches the shape onto which the map is projected. Thus, it is along the point or line(s) of tangency that map distortion will be at a minimum. The Mercator projection is a prime example as the least distortion occurs along the equator or line of tangency, while much greater distortion occurs away from the line of tangency, near the poles^{[32]}.
Tangent and secant are the two types of cases of tangency in cartography. In the tangent case, there is only one point or line of tangency  always occurring on the surface of the globe. A point of tangency occurs if the globe projects onto a flat plane that contacts the globe at only one point^{[33]}. A line of tangency occurs if the globe projects onto a 3D shape such as a cylinder or a cone from inside, while the globe only contacts the shape along one line. The secant case involves inserting the point or line of tangency into the globe instead of remaining on the surface. Thus, a point of tangency inside the globe results in a line of tangency on the surface (secant planar projections), and a line of tangency inside the globe results in two lines of tangency on the surface (secant conic or secant cylindrical projections)^{[34]}. Secant projections lessen distortion because the globe makes contact with the plane along a line instead of at one point, or makes contact with a 3D shape along two lines instead of only one.
Oblique Projection
The oblique projection is used to align an area on a map so that the map has north oriented in any direction on the map other than 0, 90, 180, or 270 degrees. In other words North can be not be exactly on the top, bottom, left or right side of the map. This type of map orientation is used in many instances to show an area in a more comprehensive way. An example of this would be to show the entire area of Japan. One of the easiest ways to show the entire extent of Japan on a map is to align the country longways going left to right on the map. In order to do this North on the map would be at either a 45 or 135 degree angle to the top of the map.
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 two particular geographic locations. This is the defining characteristic an equidistant projection, such as the Azimuthal Equidistant projection or the Equirectangular projection.
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 discussion on projection categories, a sphere is assumed. However, the Earth is not exactly spherical but is closer in shape to an oblate ellipsoid, a shape which bulges around the equator. 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 smallscale 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 called a geoid, which is a complex and more or less 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 due to its complexity but is instead 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, as well as 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 the datum is intended to map. Each ellipsoid has a distinct major and minor axis. Different controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic regions (such as the North American Datum). A few modern datums, such as WGS84 (the one used in the Global Positioning System GPS), are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of some accuracy in smaller regions.
References
 ↑ Kimmerling, A. John; Buckley, Aileen R.; Muehrcke, Phillip C.; Muehrcke, Juliana O. (2012). Map Use: Reading, Analysis, Interpretation. EsriPress Academic. 35.
 ↑ BHATIA, AATISH. "How a 19th Century Math Genius Taught Us the Best Way to Hold a Pizza Slice." Wired.com. Conde Nast Digital, 5 Sept. 2014. Web. 20 Sept. 2015.; [1]
 ↑ Tyner, Judith, Principles of Map Design, Guilford Press, 2010, 51; [1] Buckley, Aileen, "Fonts in ArcGIS Symbols," ESRI, 2012 Accessed 15 October 2012
 ↑ http://www.geo.hunter.cuny.edu/~jochen/gtech201/lectures/lec6concepts/map%20coordinate%20systems/how%20to%20choose%20a%20projection.htm
 ↑ Map Projections
 ↑ Sinusoidal Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ http://www.progonos.com/furuti/MapProj/Normal/ProjPCyl/projPCyl.html
 ↑ Peter H. Dana, Map Projection Overview Data retrieved 10/13/12Map Projection Overview,Peter H. Dana,The Geopgrapher's Craft Project, Department of Geography, The University of Colorado at Boulder. Accessed 13 October 2012
 ↑ Clarke, Keith C. (2009). Thematic Cartography and Geovisualization. Pearson Prentice Hall, Upper Saddle River, NJ 07458. 138139.
 ↑ http://www.progonos.com/furuti/MapProj/Normal/ProjCon/projCon.html
 ↑ http://www.gapsystem.org/~history/Projects/Hoyer/S5.html
 ↑ Hunter College of the City University of New York. http://www.geography.hunter.cuny.edu/~jochen/GTECH361/lectures/lecture04/concepts/Map%20coordinate%20systems/Classifying%20conic%20and%20pseudoconic%20projections.htm. Retrieved 21 Oct 2013
 ↑ An Album of Map Projections (US Geological Survey Professional Paper 1453), John P. Snyder & Philip M. Voxland, 1989, p. 4.
 ↑ Gnomonic Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ Gnomonic Projection. John Savard website, http://members.shaw.ca/quadibloc/ Accessed 18 November 2005.
 ↑ Orthographic Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ Stereographic Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ Equidistant Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ Lambert Azimuthal EqualArea Projection. From MathWorldA Wolfram Web Resource. Accessed 24 May 2010.
 ↑ Enlarging the Heart of a Map. Matching the Map Projection to the Need, Chapter 6, Figure 65; Cartography and Geographic Information Society publication. Accessed 24 May 2010.
 ↑ Map Projections
 ↑ Cite error: Invalid
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 ↑ Lo, C.P. and Albert K. W. Yeung (2002). Concepts and Techniques of Geographic Information Systems, Upper Saddle River NJ: Pearson Prentice Hall
 ↑ Lo, C.P. and Albert K. W. Yeung (2002). Concepts and Techniques of Geographic Information Systems, Upper Saddle River NJ: Pearson Prentice Hall
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 ↑ Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Review 7, 190200, 1944.
 ↑ ^{27.0} ^{27.1} ^{27.2} ^{27.3} Data Projections. Geo Community Web site. Accessed 18 May 2010.
 ↑ ^{28.0} ^{28.1} ^{28.2} ^{28.3} ^{28.4} ^{28.5} Peter H. Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder
 ↑ ^{29.0} ^{29.1} Weisstein, Eric W. Map Projections. From MathWorldA Wolfram Web Resource. Accessed 18 May 2010
 ↑ The HammerAitoff Projection
 ↑ Judith A. Tyner, Principles of Map Design, The Guilford Press, 2010, p. 118. ISBN 9781606235447
 ↑ Mercator projection. Retrieved September 23, 2017, from http://www.geography.hunter.cuny.edu/~jochen/GTECH361/lectures/lecture04/concepts/Map%20coordinate%20systems/Mercator%20projection.htm
 ↑ Point of Tangency: Definition & Example. Retrieved September 23, 2017, from http://study.com/academy/lesson/pointoftangencydefinitionexample.html
 ↑ Map Projections  types and distortion patterns. Retrieved September 23, 2017, from http://geokov.com/education/mapprojection.aspx
 Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002
 Snyder, J.P., Album of Map Projections, United States Geological Survey Professional Paper 1453, United States Government Printing Office, 1989.
 Snyder, John P. (1987). Map Projections  A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C.. This paper can be downloaded from USGS pages
 Paul Anderson's Gallery of Map Projections  PDF versions of numerous projections, created and released into the Public Domain by Paul B. Anderson, member of the International Cartographic Association's Commission on Map Projections"
External links
 Shapefile Projection Finder  Detect the projection of a shapefile
 University of Colorado at Boulder  Map Projection Overview with Illustrations
 USGS Map Projections Info
 What are map projections? ESRI ArcGIS online help. Accessed 18 May 2010.
 Weisstein, Eric. Map Projections. From MathWorldA Wolfram Web Resource. Accessed 18 May 2010
 A Cornucopia of Map Projections  A visualization of distortion on a vast array of map projections in a single image.
 G.Projector, free software by NASA GISS can render many projections.
 Map Projections. The world we live in... HyperMaths.org: Sorted list and descriptions
 RadicalCartography.net: Table of examples and properties of all common projections
 UFF.br: An interactive JAVA applet to study deformations (area, distance and angle) of map projections
 US Geological Survey overview
 Deetz, Charles H.; Adams, Oscar S. Elements of Map Projection with Applications to Map and Chart Construction. United States Coast and Geodetic Survey Special Publication No. 68 (1934).
 USGS Map Projections: A Working Manual, freely downloadable book by USGS with details on most projections, including formulas and sample calculations.
 Map projections intro
 MathWorld's formulae
 Progonos.com: How Projections Work
 PDFs of projections
 Mapthematics: GIFs of projections
 U.S. WWII Newsmap, "Maps are Not True for All Purposes, These are three of many projections", hosted by the UNT Libraries Digital Collections
 BTInternet: Java applet for interactive projections
 3DSoftware: USGS info
 Geodesy, Cartography and Map Reading from Colorado State University
 MapRef: A collection of map projections and reference systems for Europe
 KartoWeb: What is a map projection?
 NewMag: The World Turned Upside Down by Katy Kramer
 PROJ.4 MapTools: Cartographic projections library
 GMT (Generic Mapping Tools), for creating maps, processing data, and learning firsthand about projections
 Understanding Map ProjectionsPDF (1.70 MB) ESRI publication.
 World Map Projections by Stephen Wolfram based on work by YuSung Chang, Wolfram Demonstrations Project.
 B.J.S.Cahill Butterfly Map Resource Page: Octahedral Map of the World
