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How Projections Work

What lies behind a projection? Which rules tell the cartographer where coastlines are plotted? How can one define a mapping (mathematically, a conversion between two sets of values) from sphere coordinates to planar points? How good is that transformation?

“Projective”/Perspective/Geometric vs. “Algorithmic” Projections

Some projections of the azimuthal, cylindrical and conic families have a direct geometric interpretation as light rays projected from a source intercept the Earth and, according to laws of perspective, “draw” its features on a surface. The latter may be a plane, yielding the map itself, or an intermediate shape like a cylindrical or conical shell.

On the other hand, many projections are only distantly inspired by geometric principles. For instance, Mercator's cylindrical projection can't be visualized as a perspective process unless:

In all three cases the complexity negates the usefulness of a perspective model.

Indeed, many projections have simply no geometric or physical interpretation, and are described purely by mathematical formulae. I.e., the cartographer devises a spherical-to-flat mapping according to some desirable but arbitrary property or constraint.

Forward and Inverse Formulas

Perspective or not, a projection can be defined by two sets of mapping equations:

Forward and inverse mapping convert between coordinates

Coordinate transformations defined by mapping relations

Usually those relations are not functions, as the same point on the sphere may be represented by several points on the map. Instead of Cartesian distances, plane polar coordinates (radius rho, angle theta) can be used, being in fact easier to express for many projections.

Although not generally presented here, inverse mapping makes possible calculating the geographic location given a point on a map or an aerial/satellite photograph. Thus, it is relevant to several problems, like interactive mapping applications which, when the user clicks on the map, respond depending on which building, street, city or whatever is georeferenced there. It is of course important for reprojecting, i.e., converting an already projected map to other projections, and translating between different geographic databases.

Deriving Projection Formulae

Even those without an interest in mathematics could get a fresh insight on the geographical sciences by understanding a projection formula or two; however, the reader can instead skip ahead to the main projection groups.

The next sections sketch the actual process for deriving mapping formulae for a few projections:

The shape of the map is defined beforehand for all but the Winkel, Kavrayskiy VII and (depending on the approach) Sanson-Flamsteed projections, where it is a consequence of the projections's constraints. Only Mollweide's, Craster's, azimuthal equal-area and Winkel derivations require basic calculus, numerical methods, or both.

Some Conventions

Instead of commonplace degrees, minutes and seconds of arc, in cartographic mathematics any angles including latitude and longitude are more usefully measured in radians, since the length of a circular arc — thus distances along a great circle — can be directly calculated by its radius multiplied by the angle in radians. E.g., a straight angle of 180° is equivalent to pi radians; all points at latitudes 60°N and 60°S are R pi / 3 units away from the Equator. Northern latitudes and eastern longitudes are arbitrarily considered positive angles; e.g., 45°S is expressed as -pi/4.

Both forward and inverse mapping require a scaling factor, which determines but must not to be confused with the map's scale: while the former is a single constant, the latter unavoidably changes depending on location and direction across the map. Equations included here express the scaling factor with the constant R, which is a small non-negative fraction of the Earth's actual radius.

HomeSite MapDistortion Pattern - Tissot's IndicatricesMap Projections - ContentsAzimuthal Orthographic — August  5, 2018
Copyright © 1996, 1997 Carlos A. Furuti