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Wiechel projection

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TheWiechel projection is anpseudoazimuthal,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]

{\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 theLambert azimuthal equal-area projection.In polar representation, the required transformation is of the form

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

where $(r_{L},\theta _{L})$ and $(r_{W},\theta _{W})$ are the polar coordinates of the Lambert and Wiechel maps, respectively.Thedeterminant 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, unlikeconformal maps which follow theRiemann mapping theorem.

References

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^^a ^bMap Projections: A Reference Manual. Lev Moiseevič Bugaevskij, John Parr Snyder. 1995. p. 132.ISBN 9780748403042. Retrieved2013-02-15.

Map projection

By surface

Cylindrical

Mercator-conformal	Gauss–Krüger Transverse Mercator Oblique Mercator
Equal-area	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards
Cassini Central Equirectangular Gall stereographic Gall isographic Miller Space-oblique Mercator Web Mercator

Pseudocylindrical

Equal-area	Collignon Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Mollweide Sinusoidal Tobler hyperelliptical
Kavrayskiy VII Wagner VI Winkel I and II

Conical

Pseudoconical

Azimuthal
(planar)

General perspective	Gnomonic Orthographic Stereographic
Equidistant Lambert equal-area

Bonne	Sinusoidal Werner
Bottomley	Sinusoidal Werner
Cylindrical	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards
Tobler hyperelliptical	Collignon Mollweide
Albers Briesemeister Eckert II Eckert IV Eckert VI Equal Earth Goode homolosine Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube Strebe 1995