24.09.2011As 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 equal- area 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)
