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*A revised and expanded new edition of the definitive English work on map projections.*

## Map projection

In cartography , a map projection is a way to flatten a globe 's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties.

The study of map projections is the characterization of the distortions. There is no limit to the number of possible map projections. However, "map projection" refers specifically to a cartographic projection. Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate.

Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective.

Most of this article assumes that the surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids , whereas small objects such as asteroids often have irregular shapes.

The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid.

A model globe does not distort surface relationships the way maps do, but maps can be more useful 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 be measured to find properties of the region 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. The best known map projection is the Mercator projection. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection or the Winkel tripel projection [2] [4].

Map projections can be constructed to preserve some of these properties at the expense of others. Because the curved Earth's surface is not isometric to a plane, preservation of shapes inevitably leads to a variable scale and, consequently, non-proportional presentation of areas. Vice versa, an area-preserving projection can not be conformal , resulting in shapes and bearings distorted in most places of the map.

Each projection preserves, compromises, or approximates basic metric properties in different ways. The purpose of the map determines which projection should form the base for the map. Because many purposes exist for maps, a diversity of projections have been created to suit those purposes.

Another consideration in the configuration of a projection is its compatibility with data sets to be used on the map. Data sets are geographic information; their collection depends on the chosen datum model of the Earth.

Different datums assign slightly different coordinates to the same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection.

The slight differences in coordinate assignation between different datums is not a concern for world maps or other vast territories, where such differences get shrunk to imperceptibility. Carl Friedrich Gauss 's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion.

The same applies to other reference surfaces used as models for the Earth, such as oblate spheroids , ellipsoids and geoids. Since any map projection is a representation of one of those surfaces on a plane, all map projections distort. The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix. Many other ways have been described for characterizing distortion in projections. Rather than the original enlarged infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map.

For example, a small circle of fixed radius e. Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a bivariate map. The problem of characterizing distortion globally across areas instead of at just a single point is that it necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result.

Many have been described. Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection. 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 the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. To compare, one cannot flatten an orange peel without tearing and 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.

Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal such that the surface's axis of symmetry coincides with the Earth's axis , transverse at right angles to the Earth's axis or oblique any angle in between.

The developable surface may also be either tangent or secant to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe.

Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here. Tangent and secant lines standard lines are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a standard parallel.

The central meridian is the meridian to which the globe is rotated before projecting. 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.

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

Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid.

The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict the land surface. Auxiliary latitudes are often employed in projecting the ellipsoid. A third model is the geoid , a more complex and accurate representation of Earth's shape coincident with what mean sea level would be if there were no winds, tides, or land.

Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, conformality and equivalence. Therefore, in geoidal projections that preserve such properties, the mapped graticule would deviate from a mapped ellipsoid's graticule.

For irregular planetary bodies such as asteroids , however, sometimes models analogous to the geoid are used to project maps from. For example, Io is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. Haumea 's shape is a Jacobi ellipsoid , with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor. 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. Mercator , conic e. Albers , and plane e. Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.

Another way to classify projections is according to properties of the model they preserve. Some of the more common categories are:. Because the sphere is not a developable surface , it is impossible to construct a map projection that is both equal-area and conformal. The three developable surfaces plane, cylinder, cone provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways.

For one thing, most world projections in use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection. Lee notes,. No reference has been made in the above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding.

Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to the sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else. Lee's objection refers to the way the terms cylindrical , conic , and planar azimuthal have been abstracted in the field of map projections.

If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such a cylindrical projection for example is one which:. If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification. Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections.

But the term cylindrical as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map.

The famous Mercator projection is one in which the placement of parallels does not arise by projection; instead parallels are placed how they need to be in order to satisfy the property that a course of constant bearing is always plotted as a straight line.

## Part 2: A Projection Demo:

The magic of geographic information systems is that they bring together and associate representations from diverse sources and infer relationships based on spatial references. This ability depends on our data sources using well defined coordinate referencing systems. This is not to say that the coordinate systems need to be the same for each data source, only that the relationship between the coordinate references with some shared conception of the surface of the earth needs to be well described. Indeed, there are thousands of perfectly legitimate coordinate systems in active use. The notion of spatial referenicing systems is one of the most fundamental and interesting ideas that all users of GIS should understand. This document provides an overview of the basic ideas. Latitude and Longitude provide a framework for referencing places on the earth.

The shape of the earth is roughly spherical wheres as maps are two dimensional. Map projection is a set of techniques designed to depict with reasonable accuracy the spherical earth in a two-dimensional i. Map projection types are created by an imaginary source of light projected inside the earth. Common Map Projections. Exploring Map Projections Created using D3, Map Projection Transitions provides an excellent way to visualize a wide range of map projections. What is a Map Projection? How is the 3D surface of the earth modeled onto a 2D surface?

In cartography , a map projection is a way to flatten a globe 's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is the characterization of the distortions. There is no limit to the number of possible map projections. However, "map projection" refers specifically to a cartographic projection. Despite the name's literal meaning, projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate.

## Coordinate Systems and Map Projections

GIS data differs from other data types, primarily because it contains geographic coordinates describing the location of the data on the earth. Registration Policy. GDC Registration.

*There are several ways to refer to a coordinate system. Some people casually refer to any coordinate system as a "projection", but this is not strictly true.*

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A map projection is a systematic rendering of locations from the curved Earth surface onto a flat map surface. Nearly all projections are applied via exact or iterated.

Geographic Coordinates Systems & Map Projections. B.1 Approximating the Earth's Shape. To display maps, geographically referenced data located on the.

Geographic Coordinate System. Approximation of the Earth. Datum. Box NGS BenchMark Database. Map Projections. Box How to.

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