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# Map projection

A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a two-dimensional plane. This process is typically a mathematical procedure. Some methods are based on graphical, or geometric procedures, but in the end any projection can be expressed mathematically.

The creation of a map projection involves three steps, the first two in which information is lost:

1. selection of a model for the shape of the earth or planetary body (usually choosing between a sphere or ellipsoid)
2. transform geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y).
3. reduce the scale (in manual cartography this step came second, in digital cartography it comes last)

 Table of contents 1 Metric properties of maps 2 Choosing a projection surface 3 Orientation of the projection 4 Using globes vs. projecting on a plane 5 Choosing a model for the shape of the Earth 6 Categories 7 References 8 Links

## Metric properties of maps

Maps assume that the viewer has an orthogonal view of the map (they are looking straight down on every point). This is also called a perpendicular view or normal view. Some metric properties of a map are
Most map projections, both practically and theoretically, 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 is helpful to think of a globe with a light source placed at some definite point with respect 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

If a surface can be transformed onto another surface without stretching, tearing, or shrinking, then the surface is said to be an applicable surface. The sphere and ellipsoid are not applicable with a plane surface, so any projection that attempts to project them on a flat sheet will have to distort the image (similar to the impossibility of making a flat sheet from an orange peel). A surface that can be unfolded or unrolled into a flat 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 since they can be unfolded into a flat sheet without distorting the projected image (although the original projection of the earth's surface on the cylinder or cone would be distorted).

## Orientation of the projection

Once a choice is made between using a cylinder or cone is made, the orientation for that shape must be chosen (how the cylinder or cone is "placed" on the earth). The orientation of the projection surface can be normal (inline with the earth's axis), transverse (at right angles to the earth's axis) or oblique (any angle in between). These surfaces may also be either tangent or secant to the sphere or ellipsoid (if you see both a 1st and 2nd parallel on a projected map then the projection must be secant).

## Using globes vs. projecting on a plane

The globe is the only way to represent the earth without distorting one or more of the above-mentioned metric properties. Globes have the advantage of being true to metric properties and able to provide a true picture of spatial relationships on the earth's surface. The disadvantages of the globe are that it is impractical to make large-scale maps with it, it is difficult to measure on a globe, one can't see the whole world at once and it is difficult to handle and transport a globe (unlike a folding map).

The flat map has the disadvantage of always distorting one or more of the metric properties and it is more difficult to get a true picture of the spatial relationships between objects. Flat maps have numerous advantages, however: it is not practical to make large or even medium scale globes, it is easier to measure on a flat map, easy to carry around, and one can see the whole world at once.

Scale in particular is affected by the choice between using a globe vs. a plane. Only a globe can have a constant scale throughout the entire map surface. The scale for flat maps will vary from point to point and may also vary in different directions from a single point (as in Azimuthal maps). The scale for a flat map can only be true at specific points or along specific paths, and never across areas of any extent. The 'scale factor is therefore used to measure the difference between the idealized scale and the actual scale at a particular point on the map and in a particular direction at that point.

## Choosing a model for the shape of the Earth

The projection is also affected by how the shape of the earth is approximated. In the following discussion on projection categories, a sphere is assumed, but 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 (features are small) such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to jusify 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. 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 going to be used for. Each ellipsoid has a distinct major and minor axis and different controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized and used for specific geographic regions (such as the North American Datum). A few modern datums, such as 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.

## Categories

A fundamental projection classification is based on 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 classified as cylindrical (e.g., Mercator projection), conic (e.g., Albers projection), azimuthal or plane (e.g., stereographic projection).

There are several different types of projections that aim to accomplish different goals while sacrificing data in other areas through distortion.

• Area-preserving, called equal-area or equiareal or equivalent or authalic
• Shape-preserving, called conformal or orthomorphic
• Direction preserving, called azimuthal (but only possible from the central point)
• Distance preserving - equidistant (preserving distances between one or two points and every other point)
NOTE: It is impossible to construct a map projection that is both equal-area and conformal.

The two major concerns that drive the choice for a projection are the compatibility of different data sets and the amount of tolerable metric distortions. On small areas (large scale) data compatibility issues are more 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.

### Organized by surface

#### Cylindrical projections

Cylindrical projections are constructed by wrapping a cylinder around the Earth, projecting

##### Pseudo-Cylindrical Projections

Pseudo-cylindrical projections are created mathematically, representing the central meridian and each parallel as a straight line. Each pesudo-cylindrical 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 Projection
• Molleweide Projection
• Goode's Homolosine
• Eckert IV and Eckert VI

#### Conic Projections

• Equidistant Conic Projection
• Lambert Conformal Conic Projection
• Albers Conic Projection

##### Pseudo-conic projection

• Bonne Projection
• Werner cordiform projection 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 Projection

#### Azimuthal projections

Azimuthal projections touch the earth to a plane at one tangent point; angles from that tangent point are preserved, and distances from that point are computed by a function independent of the angle.

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

• Azimuthal equidistant projection 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 equal to surface distance on the earth.
• Lambert Azimuthal equal-area projection. Distance from the tangent point on the map is equal to straight-line distance through the earth.
• Azimuthal conformal projection is the same as stereographic projection. It can be constructed by using the tangent point's antipode as the point of perspective.
• 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.
• Gnomonic Projection displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth.
• Logarithmic Azimuthal Projection 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.

### Organized by Preservation of a metric property

#### Conformal projections

Conformal map projections preserve angles locally.

#### Equal-area projections

These projections preserve area.

• Gall Cylindrical Equal-Area Projection
• Albers Conic Projection
• Lambert Azimuthal Equal-Area Projection
• Mollweide Projection
• Briesemeister Projection
• Sinusoidal Projection
• Goode's Homolosine

#### Equidistant projections

These preserve distance from some standard point or line.

• Plate Caree
• Azimuthal Equidistant
• Equidistant Conic
• Sinusoidal Projection
• Werner Cordiform

#### 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".

• Robinson Projection
• Van der Grinten Projection
• Miller Cylindrical Projection
• Winkel Tripel Projection

### Other noteworthy Projections

• Dymaxion Projection

## 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.
• Synder, J.P., Map Projections - A Working Manual, United States Geological Survey Professional Paper 1395, United States Government Printing Office, 1987.