The distortion is a measure for the geometrical deformation which occurs at maps while mapping the spherical Earth's surface onto a plain area. The distortion is a criterion for the quality of maps.
In consequence of the THEOREMA EGREGIUM by C.F. Gauss it is impossible to construct maps which are an accurate image of the Earth's surface. Single curves however can be projected while preserving length.
What we want is to keep distortion in tolerable limits.
In case of a projection of an area A to an area A*, two main directories of distortion exist at any point. These directories are orthographic to each other. We named the two main distortions at a point of area a and b . They are also called the main scales of a projection. a and b are defined by the Gauss' Fundamental Values of the areas A and A* (cp. for example [9] under literature).
To visualise the distortion we make use of Tissot's Indicatrix.
In doing so we consider the image of a infinitely small circle with radius 1 on the pre-image (the sphere or the rotation ellipsoid). The image of the mentioned circle appears as an ellipse with major semi-axis a and minor semi-axis b (see figure). Tissot's Indicatrix is also called ellipse of distortion.
For length-preserving projections we have: | a = 1 or b = 1 . |
For conformal projections we have: | a = b . |
For equal-area projections we have: | a * b = 1 . |
In addition to the main scales a and b as the maximal and minimal scale error in a chosen point Q we define the scale errors h and k in Q (cp. for example [9] under literature): h is the scale error along the meridian and k the scale error along the parallel of latitude through Q.
If the parameter lines of the image area A* intersect at right angles (e.g. for the true projections) h and k are equal to a and b, thus:
in case of h > k there is a = h and b = k ,
in case of k > h there is a = k and b = k .
See also | |
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Derivation of cylindrical projections |