Map projection

A map projection is a geometric function that transforms the earth's curved, ellipsoidal surface onto a flat, 2-dimensional plane. 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.

By nature, all map projections distort different properties of the spheroid-based features. Different projections exist in order to preserve certain features but at the expense of others. Depending on the purpose of the map, some distortions are acceptable while others are not.

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.

=Background=

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 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:


 * Area
 * Shape
 * Direction
 * Distance

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 or message. A proper projection helps the viewer correctly interpret the information contained on the map, while the misuse of projection (either intentionally or unintentionally) gives viewers a skewed perspective of the area or information they are viewing. Poor use of projection can have a variety of negative consequences, such as the viewer gaining an inaccurate mental perception of an area or incorrect understanding of the map's message, or a user becoming lost. Maps are created for a variety of purposes, and as a result, there is no single map projection that is better than the rest. Therefore, when choosing a map projection, it is useful to consider a number of factors such as the following:
 * Scale: choosing the map projection depends largely on the scale of the map, and projection choice is more important for small-scale maps (those that cover large areas) such as regions or continents. Use of a suitable projection for world maps is particularly important. For example, distortion in a Mercator projection for a large scale map of city streets would not be nearly as noticeable as the distortion of landmasses present in a Mercator world map.
 * Size and Shape of the Area: different projections should be chosen for areas on a North-South versus East-West orientation. For example, a map showing Chile (North-South orientation) would be made with a different projection than a map showing the continental United States, or Western Europe (East- West orientations).
 * Location: longitudinal location of the map area determines what type of projection should be used (for example, a conic projection for areas in the mid-latitudes with east- west extent, an azimuthal projection for displaying polar areas, etc.).
 * Purpose: the intended use of a map should be tied directly to its projection. Before choosing a projection, it is important to consider the objectives of the map user. Possible uses such as navigation, research, and general reference will require different projections. For example, a Mercator projection is best suited for maps intended to be used for navigation, while a Robinson or Mollweide projection is better for general reference world maps.
 * Audience: the sophistication of the map user should be considered when determining the complexity of the projection. For example, an interrupted projection may not be appropriate for young map users because it could greatly confuse them regarding the nature of the earth. In this case, a better option for a younger audience would be the Robinson projection, since it is more intuitive and simplistic.

=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:

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

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.

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. 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 north-south in extent.



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.

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.
 * 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.

Other Projections:

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.

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. 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.

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 non-conformal projection. This projection would be useful in an educational setting where learning about the shape of continents or countries and recognizing them is more important than accurate size. However, understand that shape is not perfectly preserved in this type of 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
 * Roussilhe
 * Lambert conformal conic
 * Quincuncial map
 * Guyou hemisphere-in-a-square projection

Equal-area


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. This projection would be useful in an academic setting where the map is used to measure or compare area. The following projections preserve area:


 * Gall orthographic (also known as Gall-Peters, or Peters, projection)
 * Albers equal-area conic
 * Lambert azimuthal equal-area
 * Mollweide projection
 * Hammer
 * Briesemeister
 * Sinusoidal
 * Werner
 * Bonne
 * Bottomley
 * Goode homolosine
 * Hobo-Dyer
 * 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. 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
 * Two-point equidistant: two "control points" are arbitrarily chosen by the map maker. Distance from any point on the map to each control point is proportional to surface distance on the earth.

Gnomonic


Projected using the center of the Earth as an origin. Great circles are displayed as straight lines:
 * Gnomonic projection

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


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

Compromise projections


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
=Construction of a map projection= The creation of a map projection involves three steps:


 * 1) Selection of a model for the shape of the Earth or planetary body (usually choosing between a sphere or ellipsoid). Because the Earth's actual shape is irregular, information is lost in this step.
 * 2) Transformation of geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y)
 * 3) 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.

Latitude may have a significant impact on choosing which projection that should be used. Through the distortion in the polar regions on typical map projections, mapping may be difficult in higher latitudes and the projection type becomes an important factor. Selecting a projection such as a conic or polar azimuth enable us to accurately map these regions.

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.

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. A line of tangency occurs if the globe projects onto a 3-D 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). 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 3-D 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 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.

=References=
 * 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.
 * 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"