Map Projections

Geographer is a jack of all trades and master of none” Map projection A map projection is any method used in cartography to represent the two-dimensional curved surface of the earth or other body on a plane. The term "projection" here refers to any function defined on the earth's surface and with values on the plane, and not necessarily a geometric projection. Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. One can see this mathematically as a consequence of Gauss's Theorema Egregium. Flat maps can be more useful than globes 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 facilitate measuring properties of the terrain 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 flat maps motivate the development of map projections. Metric properties of maps Many properties can be measured on the earth's surface independently of its geography. Some of these properties are: ● Area ● Shape ● Direction ● Bearing ● Distance ● Scale Map projections can be constructed to preserve one or more of these properties, though not all of them simultaneously. Each projection preserves or compromises or approximates basic metric properties in different ways. The purpose of the map, then, determines which projection should form the base for the map. Since many purposes exist for maps, so do many projections exist upon which to construct them. Another major concern that drives the choice of a projection is the compatibility of data sets. Data sets are geographic information. As such, their collection depends on the chosen model of the earth. Different models assign slightly different coordinates to the same location, so it is important that the model be known and that chosen projection be compatible with that model. 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. Construction of a map projection The creation of a map projection involves three steps: ● Selection of a model for the shape of the earth or planetary body (usually choosing

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Geographer is a jack of all trades and master of none” ● ●

between a sphere or ellipsoid) Transformation of geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y) Reduction of the scale (it does not matter in what order the second and third steps are performed)

Because the real earth's shape is irregular, information is lost in the first step, in which an approximating, regular model is chosen. Reducing the scale may be considered to be part of transforming geographic coordinates to plane coordinates. 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 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. The sphere and ellipsoid are not developable surfaces. Any projection that attempts to project a sphere (or an ellipsoid) on a flat sheet will have to distort the image (similar to the impossibility of making a flat sheet from an orange peel). One way of describing a projection is to project first from the earth's surface to a developable surface such as a cylinder or cone, followed by the simple second step of unrolling 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. Orientation of the projection Once a choice is made between projecting onto a cylinder, cone, or plane, the orientation of the shape must be chosen. The orientation is how the shape is placed with respect to the globe. 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 spherical or ellipsoidal globe. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Insofar as preserving metric properties go, it is never advantageous to move the developable surface away from contact with the globe, so that practice is not discussed here. Scale 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.

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Geographer is a jack of all trades and master of none” Some possible properties are: The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a conformal map. Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect. Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the Mercator projection in normal aspect. Scale is constant along all straight lines radiating from two particular geographic locations. This is the defining characteristic an equidistant projection, such as the Azimuthal equidistant projection or the Equirectangular projection.

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