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.

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 ) - an
**inverse**relation performs the opposite transformation

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 , angle ) 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.

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 azimuthal orthographic projection, purely geometric, can be understood by anyone; basic trigonometry is involved only for actual computation
- Braun's and other stereographic cylindricals, arbitrary geometric projections
- Kavrayskiy's VII, a concisely described compromise projection
- the Sanson-Flamsteed, also called sinusoidal, a very plain and useful algorithmic projection demanding only simple trigonometry
- Craster's parabolic, solved using integral calculus and an interesting case of the cubic equation
- the Mollweide or Babinet, a slightly more difficult projection which requires 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.

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 radians;
all points at latitudes 60°N and 60°S are
units away from the Equator.
Northern latitudes and eastern longitudes are arbitrarily
considered positive angles; e.g., 45°S is expressed as .

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 ,
which is a small non-negative fraction of the Earth's actual radius.

Copyright © 1996, 1997 Carlos A. Furuti