Map Projections: How Projections Work

Map Projections

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

Imaginary development of orthographic projection

Conceptual model of a pure perspective projection. Light rays pierce the Earth drawing a map on a flat square suspended in space. Notice how points correspond on Earth and map.

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:

forward or direct relation converts polar coordinates (longitude , latitude , Earth radius R) to Cartesian coordinates (horizontal distance x from origin, vertical distance y), provided a convenient scaling factor (not to be confused with the map scale). Equations included here assume a unitary scaling factor.
inverse relation performs the opposite transformation

Coordinate transformations defined by mapping relations

Usually those relations are not functions (e.g., 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 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 converting an already projected map to other projections.

Deriving Projection Formulae

Even those without a mathematical background 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 algorithm derivation
the Braun stereographic cylindrical, an arbitrary geometric projection
the Sanson-Flamsteed, also called sinusoidal, a very plain algorithmic projection demanding only simple trigonometry
the Mollweide or Babinet, a slightly more difficult projection solved by integral calculus and numerical analysis
two generalized 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 map shape is defined beforehand for all but the Sanson-Flamsteed and Winkel projections, where it is a consequence of the projections's constraints. Only the Mollweide, azimuthal equal-area and Winkel derivations require basic calculus, numerical methods, or both.

Instead of commonplace degrees, minutes and seconds, in cartographic mathematics all 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 / 3 units away from the Equator.
Northern latitudes and eastern longitudes are arbitrarily considered positive angles; e.g., 45°S is expressed as - / 4.