27.08.2011
As it is not possible to map from the earth
surface to a plane without distortions (intrinsics of
geometry), a lot of effort has already been done to analyse the
distortion properties. The distortions depend on the mapping
surface, its aspect and other mathematical or geometrical
properties of map projections and are a function of the position.
Even though, there may be found a specific property which is equal
for each position on the projection. In fact, many projections were
constructed by restrictions on the distortions. The methods
therefore are given by the surface theory. The following metric
distortions may be given, but the first three properties exclude
each other:

conformity or orthomorphism (locally no angular
distortion)

equivalency or authalicity (locally equalarea
properties)

partially equidistant (specific lines as meridians are
mapped with true length)

compromise or error minimised (restrictions to all
distortion properties)
The mathematical instrument to calculate
distortions is based on the Tissot Indicatrix: the first
order approximation of the mapped shape of an infinitesian small
circle on the origin surface is a ellipse, the Tissot
Indicatrix.
Tissot's
Indicatrix: distortion analysis [Voser
2003]
The analysis of this ellipse defines the
distortion properties, using the semi major axis a and the
semi minor axis b of the ellipse:

conformity: for all points, T.I. is a circle
(a=b)

equivalency: for all points, the T.I has the same area
(a*b=const)

partially equidistant (specific lines are mapped with same
length: l=const)