The Winkel tripel projection (Winkel III), a modified azimuthal

map projection In cartography, a map projection is any of a broad set of Transformation (function) , transformations employed to represent the curved two-dimensional Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In a map projection, ...

of the

world The world is the totality of entities, the whole of reality, or everything that Existence, exists. The nature of the world has been conceptualized differently in different fields. Some conceptions see the world as unique, while others talk ...

, is one of three projections proposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the

equirectangular projection The equirectangular projection (also called the equidistant cylindrical projection or la carte parallélogrammatique projection), and which includes the special case of the plate carrée projection (also called the geographic projection, lat/l ...

and the Aitoff projection: The name (

German German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also Ge ...

for 'triple') refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance.

Algorithm

\begin
 x &= \frac \left(\lambda \cos \varphi_1 + \frac\right), \\
 y &= \frac \left(\varphi + \frac\right),
\end

where ''λ'' is the longitude relative to the central meridian of the projection, ''φ'' is the latitude, ''φ'' is the standard parallel for the

, sinc is the unnormalized cardinal sine function, and :

\alpha = \arccos\left(\cos\varphi \cos \frac \right).

In his proposal, Winkel set :

\varphi_1 = \arccos \frac.

closed-form Closed form may refer to: Mathematics * Closed-form expression, a finitary expression * Closed differential form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior deri ...

inverse mapping does not exist, and computing the inverse numerically requires the use of

iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an " ...

Comparison with other projections

David M. Goldberg and J. Richard Gott III showed that the Winkel tripel fares better against several other projections analyzed against their measures of distortion, producing minimal distance, Tissot indicatrix ellipticity and area errors, and the least skew of any of the projections they studied. By a different metric, Capek's "Q", the Winkel tripel ranked ninth among a hundred map projections of the world, behind the common Eckert IV projection and

Robinson projection The Robinson projection is a map projection of a world map that shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image. The Robinson ...

s. In 1998, the Winkel tripel projection replaced the Robinson projection as the standard projection for world maps made by the

National Geographic Society The National Geographic Society, headquartered in Washington, D.C., United States, is one of the largest nonprofit scientific and educational organizations in the world. Founded in 1888, its interests include geography, archaeology, natural sc ...

. Many educational institutes and textbooks soon followed National Geographic's example in adopting the projection, most of which still utilize it.

References

External links

Table of common projections
{{Map projections Map projections

Algorithm

Comparison with other projections

See also

References

External links