# Winkel tripel projection

Winkel tripel projection

The Winkel tripel projection (Winkel III) is a modified azimuthal map projection, one of three projections proposed by Oswald Winkel in 1921. The projection is the arithmetic mean of the equirectangular projection and the Aitoff projection ["Flattening the Earth: Two Thousand Years of Map Projections", John P. Snyder, 1993, pp.231-232, ISBN 0-226-76747-7.] :

:$x = frac\left\{lambda cos\left(phi_1\right) + frac\left\{2 cos\left(phi\right)sinleft\left(frac\left\{lambda\right\}\left\{2\right\} ight\right)\right\}\left\{mathrm\left\{sinc\right\}\left(alpha\right)\left\{2\right\}$

:$y = frac\left\{phi + frac\left\{sin\left(phi\right)\right\}\left\{mathrm\left\{sinc\right\}\left(alpha\right)\left\{2\right\}$

where $lambda$ is the longitude from the central meridian of the projection, $phi$ is the latitude, $phi_1$ is the standard parallel for the equirectangular projection, and

:$alpha = arccosleft\left(cos\left(phi\right) cosleft\left(frac\left\{lambda\right\}\left\{2\right\} ight\right) ight\right),$

$mathrm\left\{sinc\right\}\left(alpha\right)$ is the unnormalized sinc function with the discontinuity removed. In his proposal, Winkel set :

:$phi_1 = arccosleft\left(frac\left\{2\right\}\left\{pi\right\} ight\right),$

Not surprisingly, a closed form inverse mapping does not exist, and computing the inverse numerically is somewhat complicated.

Goldberg & Gott show that the Winkel-Tripel is arguably the best overall whole-earth map projection known, producing very small distance errors, small combinations of ellipticity and area errors, and the smallest skewness of any map. [" [http://www.physics.drexel.edu/~goldberg/projections/goldberg_gott.pdf Large-Scale Distortions in Map Projections] ", 2007, David M. Goldberg & J. Richard Gott III, 2007, V42 N4.]

In 1998, the Winkel Tripel projection replaced the Robinson projection as the standard projection for world maps made by the National Geographic Society. Many educational institutes and textbooks followed National Geographic's example in adopting the projection, and most of those still use it.

References

* [http://welcome.warnercnr.colostate.edu/class_info/nr502/lg2/projection_descriptions/winkel_tripel.html Winkel Tripel projection] , from "Geodesy, Cartography and Map Reading" (Denis J. Dean, Colorado State University)
* [http://www.radicalcartography.net/?projectionref Table of common projections]

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