Wiechel projection explained

The Wiechel projection is an pseudoazimuthal, equal-area map projection, and a novelty map presented by William H. Wiechel in 1879. When centered on the pole, it has semicircular meridians arranged in a pinwheel. Distortion of direction, shape, and distance is considerable in the edges.^[1]

In polar aspect, the Wiechel projection can be expressed as so:

\begin{align}x & = R \left( \sin λ \cos \phi - \left( 1 - \sin \phi \right) \cos λ \right), \\ y & = - R \left( \cos λ \cos \phi + \left( 1 - \sin \phi \right) \sin λ \right). \end{align}

The Wiechel can be obtained via an area-preserving polar transformation of the Lambert azimuthal equal-area projection.In polar representation, the required transformation is of the form

\begin{align}r _{W} & = r _{L,} \\ \theta _{W} & = \theta _{L} - (1)/(2) \arcsin r _{L, \end{align}}

where

(r _{L,\theta} _{L} ) and (r _{W,\theta} _{W)} are the polar coordinates of the Lambert and Wiechel maps, respectively.The determinant of the Jacobian of the transformation is equal to unity, ensuring that it is area-preserving.The Wiechel map thus serves as a simple example that equal-area projections of the sphere onto the disk are not unique, unlike conformal maps which follow the Riemann mapping theorem.

See also

• Lambert azimuthal equal-area projection

Notes and References

1. Book: Map Projections: A Reference Manual . 2013-02-15 . 1995 . Lev Moiseevič Bugaevskij, John Parr Snyder . 9780748403042 . 132.