Wiechel projection

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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:^[1]

${\displaystyle {\begin{aligned}x&=R\left(\sin \lambda \cos \phi -\left(1-\sin \phi \right)\cos \lambda \right),\\y&=-R\left(\cos \lambda \cos \phi +\left(1-\sin \phi \right)\sin \lambda \right).\end{aligned}}}$

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

${\displaystyle {\begin{aligned}r_{W}&=r_{L},\\\theta _{W}&=\theta _{L}-{\frac {1}{2}}\arcsin r_{L},\end{aligned}}}$

where ${\displaystyle (r_{L},\theta _{L})}$ and ${\displaystyle (r_{W},\theta _{W})}$ 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.