Abstract
There has been a dramatic increase in the handling of geospatial information, and also in the production of maps. However, because the Earth is three-dimensional, geo-referenced data must be projected on a two-dimensional surface. Depending on the area being mapped, the projection process generates a varying amount of distortion, especially for continental and world maps. Geospatial users have a wide variety of projections too choose from; it is therefore important to understand distortion characteristics for each of them. This chapter reviews foundations of map projection, such as map projection families, distortion characteristics (areal, angular, shape and distance), geometric features and special properties. The chapter ends by a discussion on projection selection and current research trends."
TopThe Map Projection Process
The Earth is essentially spherical, but is approximated by a mathematical figure –a datum surface. For the purpose of world maps, a sphere with radius RE = 6371km is a satisfying approximation. For large-scale maps however (i.e., at the continental and country scale), the non-spherical shape of the Earth is represented by an ellipsoid with major axis a and minor axis b. The values of a and b vary with the location of the area to be mapped and are calculated in such a way that the ellipsoid fits the geoid almost perfectly. The full sized sphere is greatly reduced to an exact model called the generating globe (see Figure 1). Nevertheless, globes have many practical drawbacks: they are difficult to reproduce, cumbersome for measuring distances, and less than the globe is visible at once. Those disadvantages are eliminated during the map projection process, by converting the longitude and latitude angles (λ,
) to Cartesian coordinates (Canters and Decleir 1989):
,
(1)Figure 1. The map projection process: the sphere, approximated by a mathematical figure is reduced to a generating globe that is projected on a flat surface. (after Canters and Decleir 1989)
TopClassification Of Map Projections By Families
Three major projections classes are named after the developable surface onto which most of the map projections are at least partially geometrically projected. All three have either a linear or punctual contact with the sphere: they are the cylindrical, conical and azimuthal. The advantage of these shapes is that, because their curvature is in one dimension only, they can be flattened to a plane without any further distortion (Iliffe, 2000). The pseudocylindrical, pseudoconic, pseudoazimuthal and pseudoconical projections are based on the three aforementioned families (Snyder 1987; Lee 1944).
Conical projection
When a cone wrapped around the globe is cut along a meridian, a conical projection results. The cone has its peak -also called apex- above one of the two Earth's poles and touches the sphere along one parallel of latitude (Figure 2a). When unwrapped, meridians become straight lines converging to the apex, and parallels are represented by arcs of circle. The pole is either represented as a point or as a line. Their spacing along the meridians is defined to meet desired properties. When the cone is secant to the globe, it bisects the surface at two lines of latitude (Figure 2b).
Figure 2. Illustration of the tangent conical projection in (a) and a secant projection in (b). Illustration of the tangent cylindrical projection in (c) and its secant counterpart in (d). Distortion is minimum on the contact lines and increases away from those parallels of latitude.
Key Terms in this Chapter
Latitude: The angle between the point and the equator along a meridian (Figure 1)
Parallel of Latitude: denoted f, a parallel is formed by circles surrounding the Earth and parallel to the Equator. Parallels of latitude are drawn equally spaced within the 90º separation between the poles ant the Equator. The circles are numbered from 0º at the Equator to 90º at the poles. The radius of parallel decreases polewards at a rate of .
Longitude: The angle on the equatorial plane between the meridian of the point and the central meridian (through Greenwich, England)
Great Circle: An imaginary circle made on the Earth’s surface by a plane passing through its center. It is also the shortest distance between two points on the Earth. Represented by straight on a gnomonic projection.
Cartesian Coordinate System: A plane coordinate system based on a map projection, where the location of a feature is determined by x and y.
Central Meridian: An imaginary meridian that serves as the starting point for measuring longitude. Passes through the Royal Observatory in Greenwich, England
Rhumb Line: Lines of constant bearing (direction). Represented as a straight line on the Mercator projection, but as a curve on the gnomonic.
Ellipsoid: a model that approximated the shape of the Earth. Also called spheroid.
Orthophanic: A projection is said orthophanic when it is right-appearing. The Robinson projection, is an example. Such projections are pleasing to the eye, especially because the distortion of continental shapes remains acceptable from a perceptive point of view (McLeary 1989).
Generating Globe: Also called reference globe, it is a reduced model of the Earth, from which map projections are made.
Meridians of Longitude: denoted ?, meridians connect the North Pole to the South poles by a set of imaginary lines, and perpendicular with each parallels of latitude. The length of 1º of longitude varies with varying latitude. At the Equator, they are the same length as a degree of latitude, but decrease polewards (Snyder 1987)
Geographic Coordinate System: A location reference system for spatial features on the Earth, where the location of a geographic feature is determined by angles ? and.
Transformation Formula: Mathematical formula allowing the transfer of data from geographic coordinates to Cartesian coordinates.