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intrinsics of map projections Map Projection
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 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)