Map Projection

A map projection is one of many methods used to represent the 3-dimensional surface of the earth or other round body on a 2-dimensional plane in cartography (mapmaking). This process is typically, but not necessarily, a mathematical procedure (some methods are graphically based).

The creation of a map projection involves three steps in which information is lost in each step:

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

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. The metric properties or a map are

• area
• shape
• direction
• distance
• scale

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 or 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 around (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 effected by the choice between using a globe vs. a plane. Only a globe can have a constant scale throughout the entire map surface and 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 along one or two lines or points (tangent or secant points/lines). 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.

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 ellipsoid with a bulge around the equator. Selecting a model for a shape of the earth involves a choice between the advantages and disadvantages between using a sphere vs. 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). Datums are always 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).

Categories
Projection classification is based on type of projection surface that is used. The projections are described in terms of placing a gigantic planar surface in contact with the earth, followed by an implied scaling operation. These surfaces are classified as cylindrical (exm. Mercator projection), conic (exm. Albers projection), azimuthal or plane (polar region projections).
There are several different types of projections that aim to accomplish different goals while sacrificing data in other areas through distortion.

Area preserving projection - equal area or equivalent projection
Shape preserving - conformal, orthomorphic
Direction preserving - conformal, orthomorphic, azimuthal (only from a the central point)
Distance preserving - equidistant (shows the true distance 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.

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.
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.

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.

Azimuthal orthographic projection maps each point on the earth to the closest point on the plane.

Conformal projections
Conformal map projections preserve angles.
Mercator projection wraps a cylinder around the earth; the distance from the equator on the map is

being geographical latitude, on a scale where the earth's radius is 1.

Stereographic projection touches a plane to the earth and projects each
point in a straight line from the antipode of the tangent.

Equal-area projections
These projections preserve area.
Gall-Peters projection wraps a cylinder around the earth and maps each
point on the earth to the nearest point on the cylinder.
Azimuthal equal-area: see above.
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.

Reference
Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002

More Resources:
Netlinks: Projections and Coordinate Systems