Projection

Map Projection Distortions

A globe is a useful way of representing the earth, but globes are difficult to make, awkward to carry, and hard to use for measuring areas, distances, and directions. Since there is no method of flattening a globe without distorting distance, direction, area, and shape in some way, no map projection can portray any significant portion of the earth without some distortion of these attributes. The best that can be done is to represent one of these attributes as correctly as possible at the expense of distorting the others or to distort a set of selected attributes as little as possible, while allowing more distortion in the others.

Reference Ellipsoids and Geodetic Datums

Map projections are usually based on a reference sphere or a reference ellipsoid. For spherical models, a single radius is sufficient to describe the size of the earth. For ellipsoidal models, which more closely match the gravity shape of the earth, two parameters can define the size and the shape of the threedimensional ellipse rotating in space.

A spherical or ellipsoidal globe of reduced size is the basis for a map at a specified scale. To transform shapes on the earth to a flat plane, a coordinate system defined on the earth is transformed into a coordinate system on the map.

For a spherical earth, a prime meridian and the equator will suffice to define the origin for longitude and latitude. For ellipsoidal models, a geodetic datum defines the reference ellipsoid size and shape as well as the origin and precise orientation for the lines of longitude and latitude on the earth. Some geodetic datums [Page 349]define longitude and latitude over a specific region, others for the entire globe.

Geometrical Constructions

There are three simple ways in which a flat plane can be manipulated so that shapes on the surface of the earth can be projected and transferred onto the plane. While there are many modern projections that do not fall into these simple categories, most have some relationship to these three classes:

• 1.
  A cylinder can be wrapped around the globe, with its central axis pointed north and south (regular or normal), east-west (transverse), or at some other (oblique) angle. When shapes on the globe are transferred, or projected, onto this surface, the result is a portrayal of the entire globe on a single rectangular surface. Maps showing the entire earth are often made using a cylindrical projection. The Mercator map, used for nautical charts, is an example.
• 2.
  A cone can be placed over part of the globe, such that after the projection of the globe features onto it, the cone can be unwrapped to portray part of the globe on a flat surface. Conic projections are often used for equal-area maps. The Albers Equal-Area Conic is one.
• 3.
  A flat plane can be placed on or near the globe without any curvature, so that features for a portion of the earth can be projected onto it and displayed without any wrapping or unwrapping. These planar projections are often used for equidistant or azimuthal maps showing correct angles or distances from the map center. Polar stereographic maps are planar projections.

For each of these geometric construction techniques, there are two approaches for portraying features at specified scales. The first is to design a cylinder or cone so that it touches the surface of the globe along a single circle or to design a flat plane so that it touches the globe at a single point. These tangent-projection methods result in maps in which there is a one-to-one correspondence between the scale of the globe and the scale of the map along the circle or at the single point where the map and globe touch.

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