A map projection is any orderly system of parallels and meridians on which a map can be drawn. These parallels and meridians help take real world locations and spatial references and put them in a flat plane. Any mathematical function transforming coordinates from a curved surface to a plane is considered a projection. 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. To get a simple idea of how projections are created, imagine a light bulb placed at the center of a transparent globe, then placing a piece of paper around the globe turn on the light and trace what is "projected" onto the paper. The end result is a simple map projection.
In their attempt at planar representation of actual map features such as an island or continent on the curved surface of the earth, all map projections necessarily distort some aspects of these features. 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 spheroid-based features at the expense of other properties.
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.
Carl Friedrich Gauss's Theorema Egregium 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 motivate 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 form the basis for the map. 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. Data sets are 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), on the other hand, distortion is a more important factor to consider.
 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 (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
NOTE: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.
There are also times that when working with large scales, such as districts or provinces within countries, the distortion doesn't play a significant role, in which any projection that is centered on the area of interest is okay.
 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 east-west. 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 north-south stretching (where latitude is given by φ):
- North-south stretching is equal to the 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 growing rapidly with latitude, even 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): equidistant cylindrical or plate carrée.
- North-south compression precisely the reciprocal of east-west stretching (cosine φ): equal-area cylindrical (with many named specializations such as Gall-Peters projection or Gall orthographic, Behrmann, and Lambert cylindrical equal-area). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting 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.
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.
Cylindrical projections are commonly used for world maps, regions bordering the equator, and regions that are predominantly north-south in extent.
 Pseudocylindrical projection
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.
 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 normal cylindrical projection has one point of tangency around the earth. If the cylinder circumference were to become smaller, the cylinder would pass through the earth and out the other side. This creates two secant lines, also called standard lines, on the reference globe that have no distortion at the secant lines. The use of a secant projection allows distortion to have less of a impact on certain areas of a map while at the same time increasing the distortion on other areas. For example, with a Euler conic projection the use of secant lines in the northern hemisphere decreases the area distortion in the northern hemisphere while the outer reaches on the southern hemisphere are stretched more. 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.  
 Conical  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 equal-area 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 heart-shaped.
- 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 non-concentric 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.
 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.
- 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. Works well with cognitive maps. r(d) = c ln(d/d0); locations closer than at a distance equal to the constant d0 are not shown.
Azimuthal projections are commonly used to map polar regions, roughly circular regions, or for maps of continents and hemispheres. 
 Projections by preservation of a metric property
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 non-conformal projection.
Unfortunately, however, in order to construct such a projection, the scale has to be varied considerably.
Examples of projections that are conformal include:
- Mercator - rhumb lines are represented by straight segments
- Stereographic - shape of circles is conserved
- Lambert conformal conic
- Quincuncial map
- Guyou hemisphere-in-a-square projection
A map designed around an equal-area (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. Equal-area projections are well-suited for maps of general interest and for those showing distributions over space, such as population, wildlife habitats, and land cover. An equal-area projection, however, causes distortions in shape and distance. These projections preserve area:
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. Some examples of projections that attempt to keep distances correct are:
- 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
- Sinusoidal - distances along parallels are conserved
- Werner cordiform distances from the North Pole are correct as are the curved distance on parallels
- 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.
Projected using the center of the Earth as an origin. Great circles are displayed as straight lines:
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.|
|Alaska Grid||This projection was developed to provide a conformal map of Alaska with less scale distortion than other conformal projections.|
|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.|
|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.|
|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.|
|Azimuthal Equidistant||The most significant characteristic of this projection is that both distance and direction are accurate from the central point.|
|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 star-shaped 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.|
|Bonne||This equal-area projection has true scale along the central meridian and all parallels.|
|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 north-south in extent.|
|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.|
|Craster Parabolic||This pseudo-cylindrical equal-area projection is primarily used for thematic maps of the world.|
|Cylindrical Equal Area||See Equal Area Cylindrical|
|Double Stereographic||This azimuthal projection is conformal.|
|Eckert I||This pseudocylindrical projection is used primarily as a novelty map.|
|Eckert II||A pseudocylindrical equal-area projection.|
|Eckert III||This pseudocylindrical projection is used primarily for world maps.|
|Eckert IV||This pseudocylindrical projection projection is used primarily for world maps.|
|Eckert V||This pseudocylindrical projection is used primarily for world maps.|
|Eckert VI||This equal-area projection is used primarily for world maps.|
|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.|
|Equidistant Cylindrical||One of the easiest projections to construct because it forms a grid of equal rectangles.|
|Equal Area Cylindrical||This projection is an equal-area cylindrical projection suitable for world mapping.|
|Equirectangular||This projection is very simple to construct because it forms a grid of equal rectangles.|
|Fuller||A projection of a global map onto the surface of a polyhedron, which, when expanded to a flat, two-dimensional 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.|
|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.|
|Geocentric Coordinate System||The geocentric coordinate system is not a map projection. The earth is modeled as a sphere or spheroid in a right-handed X,Y,Z system.|
|Geographic Coordinate System||The geographic coordinate system is not a map projection. The earth is modeled as a sphere or spheroid.|
|Gnomonic||This azimuthal projection uses the center of the earth as its perspective point.|
|Goode homolosine||A pseudocylindrical, equal-area map projection used for world maps. Its equal-area 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.|
|Hammer–Aitoff||The Hammer-Aitoff equal-area projection, also called the Hammer projection, is a map projection that is a modification of the Lambert azimuthal equal-area 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 equal-area projection, it is equal area, but it is no longer azimuthal.|
|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.|
|Krovak||The Krovak projection is an oblique Lambert conformal conic projection designed for the former Czechoslovakia.|
|Lambert Azimuthal Equal Area||This projection preserves the area of individual polygons while simultaneously maintaining true directions from the center.|
|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.|
|Local Cartesian Projection||This is a specialized map projection that does not take into account the curvature of the earth.|
|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.|
|McBryde–Thomas Flat-Polar Quartic||This equal-area projection is primarily used for world maps.|
|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.|
|Miller Cylindrical||This projection is similar to the Mercator projection except that the polar regions are not as distorted.|
|Mollweide||Pseudocylindrical and equal-area; 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.|
|New Zealand National Grid||This is the standard projection for large-scale maps of New Zealand.|
|Oblique Aspect Orthographic||Provides perspective views of hemispheres. Area and shape are distorted. Distances are true along the equator and other parallels.|
|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.|
|Orthographic||This perspective projection views the globe from an infinite distance. This gives the illusion of a three-dimensional globe.|
|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.|
|Peters (or Gall-Peters)||A modified equal-area cylindrical projection presented by German political propagandist Arno Peters as original, but is actually identical to the Gall projection.|
|Plate Carrée||This projection is very simple to construct because it forms a grid of equal rectangles.|
|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.|
|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 non-concentric circles. Scale is true along each parallel and along the central meridian.|
|Quartic Authalic||This pseudocylindrical equal-area projection is primarily used for thematic maps of the world.|
|Rectified Skewed Orthomorphic||This oblique cylindrical projection is provided with two options for the national coordinate systems of Malaysia and Brunei.|
|Robinson||A pseudo-cylindrical map projection which distorts shape, area, scale and distance to create attractive average projection properties.|
|Secant Projection||A secant projection contains two secant lines to increase the area of best representation.  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.|
|Sinusoidal||As a world map, this projection maintains equal area despite conformal distortion.Also known as the 'Mercator-Sanson-Flamsteed' projection.|
|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.|
|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.|
|Stereographic||This azimuthal projection is conformal.|
|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.|
|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.|
|Two-Point Equidistant||This modified planar projection shows the true distance from either of two chosen points to any other point on a map.|
|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.|
|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.|
|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.|
|Vertical Near-Side 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.|
|Winkel I||A pseudo-cylindrical projection used for world maps that averages the coordinates from the Equirectangular (Equidistant Cylindrical) and Sinusoidal projections.|
|Winkel II||A pseudo-cylindrical projection that averages the coordinates from the Equirectangular and Mollweide projections.|
|Winkel Tripel||A compromise projection used for world maps that averages the coordinates from the Equirectangular (Equidistant Cylindrical) and Aitoff projections.|
 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 (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 or line of tangency, along which there is minimal distortion.In the tangent case, the line (or point) of tangency contacts the globe at one line (or point).
The secant case is where the line (or point) of tangency is not on the surface but actually inserted into the globe. This minimizes distortion because the projection now has two points at which distortion is minimized, greatly decreasing the distortion on the edges of the map.Insofar as preserving metric properties goes, it is never advantageous to move the line of tangency away from contact with the globe.
 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.
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 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 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.
 Choosing a Proper Projection
Choosing the proper projection for a map is vital to correctly presenting the map's information. A proper projection (and noting that projection on the map) 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. 
- Scale: Projections are more important for small-scale 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 North-South versus East-West 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
- ↑ 1.0 1.1 Raisz, Erwin. General Cartography (1938), p. 81. McGraw-Hill.
- ↑ http://www.geo.hunter.cuny.edu/~jochen/gtech201/lectures/lec6concepts/map%20coordinate%20systems/how%20to%20choose%20a%20projection.htm
- ↑ Map Projections
- ↑ Sinusoidal Projection. From MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ 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. 138-139.
- ↑ http://www.progonos.com/furuti/MapProj/Normal/ProjCon/projCon.html
- ↑ http://www.gap-system.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 MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ Gnomonic Projection. John Savard website, http://members.shaw.ca/quadibloc/ Accessed 18 November 2005.
- ↑ Orthographic Projection. From MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ Stereographic Projection. From MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ Equidistant Projection. From MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ Lambert Azimuthal Equal-Area Projection. From MathWorld--A Wolfram Web Resource. Accessed 24 May 2010.
- ↑ Enlarging the Heart of a Map. Matching the Map Projection to the Need, Chapter 6, Figure 6-5; Cartography and Geographic Information Society publication. Accessed 24 May 2010.
- ↑ Map Projections
- ↑ 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
- ↑ 21.00 21.01 21.02 21.03 21.04 21.05 21.06 21.07 21.08 21.09 21.10 21.11 21.12 21.13 21.14 21.15 21.16 21.17 21.18 21.19 21.20 21.21 21.22 21.23 21.24 21.25 21.26 21.27 21.28 21.29 21.30 21.31 21.32 21.33 21.34 21.35 21.36 21.37 21.38 21.39 21.40 21.41 21.42 21.43 21.44 21.45 21.46 21.47 21.48 21.49 21.50 21.51 21.52 List of supported map projections. ArcGIS 9.3 Desktop online help. Accessed 18 May 2010.
- ↑ Lee, L. P. "The Nomenclature and Classification of Map Projections." Empire Survey Review 7, 190-200, 1944.
- ↑ 23.0 23.1 23.2 23.3 Data Projections. Geo Community Web site. Accessed 18 May 2010.
- ↑ 24.0 24.1 24.2 24.3 24.4 24.5 Peter H. Dana, The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder
- ↑ 25.0 25.1 Weisstein, Eric W. Map Projections. From MathWorld--A Wolfram Web Resource. Accessed 18 May 2010
- ↑ Judith A. Tyner, Principles of Map Design, The Guilford Press, 2010, p. 118. ISBN 9781606235447
- ↑ Tyner, Judith, Principles of Map Design, Guilford Press, 2010, 51;  Buckley, Aileen, "Fonts in ArcGIS Symbols," ESRI, 2012 Accessed 15 October 2012
- 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"
- 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 MathWorld--A 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 first-hand about projections
- PDF (1.70 MB) ESRI publication.
- World Map Projections by Stephen Wolfram based on work by Yu-Sung Chang, Wolfram Demonstrations Project.
- B.J.S.Cahill Butterfly Map Resource Page: Octahedral Map of the World