Winkel Tripel.html

A Winkel Tripel projection of the Earth

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 projection¹:

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

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

where $λ$ is the longitude from the central meridian of the projection, $φ$ is the latitude, $φ 1$ is the standard parallel for the equirectangular projection, and

$\alpha = \arccos\left(\cos(\phi) \cos\left(\frac{\lambda}{2}\right)\right)\,$

$sinc(α)$ is the unnormalized sinc function with the discontinuity removed. In his proposal, Winkel set :

$\phi_1 = \arccos\left(\frac{2}{\pi}\right)\,$

The Winkel tripel projection with Tissot's Indicatrix of deformation

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. ²

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

^ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp.231-232, ISBN 0-226-76747-7.
^ Large-Scale Distortions in Map Projections, 2007, David M. Goldberg & J. Richard Gott III, 2007, V42 N4.

External links

Winkel Tripel projection, from "Geodesy, Cartography and Map Reading" (Denis J. Dean, Colorado State University)
Table of common projections

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