Map Projections: How Projections Work

Map Projections

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How Projections Work

What lies behind a projection? Which rules tell the cartographer where coastlines are plotted? How do we set a mapping (mathematically, a conversion between two sets of values) from sphere coordinates to planar points? How good is this 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:

light rays don't follow straight trajectories, or
the light source is not a point or straight line, or
the projection surface is not a simple tube

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:

a forward or direct relation converts polar coordinates (longitude , latitude , Earth radius or its equivalent for the ellipsoidal case) to Cartesian coordinates (abscissa or horizontal distance from the origin, ordinate or vertical distance )
a inverse relation performs the opposite transformation

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. It is of course important for reprojecting, i.e., converting an already projected map to other projections.

Deriving Projection Formulae

Even those without a 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 azimuthal orthographic projection, purely geometric, can be understood by anyone; basic trigonometry is involved only for actual computation
Braun's stereographic cylindrical, an arbitrary geometric projection
Kavrayskiy's VII, a concisely described compromise projection
the Sanson-Flamsteed, also called sinusoidal, a very plain algorithmic projection demanding only simple trigonometry
Craster's parabolic, a interesting case of the cubic equation
the Mollweide or Babinet, a slightly more difficult projection solved by integral calculus and numerical analysis
two non-perspective azimuthal projections, the equidistant and equal-area, in both polar and equatorial aspects
the equidistant cylindrical, a very simple arbitrary projection, and two hybrid derivatives, Winkel I (generalized Eckert V) and II
the Aitoff, Hammer and Winkel Tripel projections, derivatives from azimuthal and cylindrical maps

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, in cartographic mathematics angles of latitude and longitude are more usefully measured in radians, since the length of a circular arc can be directly calculated by its radius multiplied by the angle in radians. E.g., a straight angle of 180° is equivalent to 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. Equations included here express the scaling factor in the constant , which usually is a small fraction of the Earth's actual radius.