Winkel tripel projection
Encyclopedia
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
, is one of three projections proposed by Oswald Winkel in 1921. The projection is the arithmetic mean of the equirectangular projection
Equirectangular projection
The equirectangular projection is a very simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100...
and the Aitoff projection
Aitoff projection
The Aitoff projection is a modified azimuthal map projection. Proposed by David A. Aitoff in 1889, it is the equatorial form of the azimuthal equidistant projection, but stretched into a 2:1 ellipse while halving the longitude from the central meridian:...
: The name Tripel (German for "triple") refers to Winkel's goal of minimizing distortion to the three factors of area, direction or distance.
where
is the longitude from the central meridian of the projection, 
is the latitude, 
is the standard parallel for the equirectangular projectionEquirectangular projection
The equirectangular projection is a very simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100...
, and
is the unnormalized cardinal sine function (with the discontinuity removed). In his proposal, Winkel set :
A closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
inverse mapping
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
does not exist, and computing the inverse numerically is somewhat complicated.
Goldberg and Gott
J. Richard Gott
John Richard Gott III is a professor of astrophysical sciences at Princeton University. He is known for developing and advocating two cosmological theories with the flavor of science fiction: Time travel and the Doomsday argument.- Exotic matter time travel theories :Paul Davies's bestseller How...
show that the Winkel tripel fares well against several other projections analyzed against their measures of distortion, producing small distance errors, small combinations of Tissot indicatrix
Tissot's Indicatrix
Tissot’s indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize distortions due to map projection...
ellipticity and area errors, and the smallest skewness
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
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 and Robinson projection
Robinson projection
The Robinson projection is a map projection of a world map, which 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....
s.
In 1998, the Winkel Tripel projection replaced the Robinson projection as the standard projection for world maps made by the National Geographic Society
National Geographic Society
The National Geographic Society , headquartered in Washington, D.C. in the United States, is one of the largest non-profit scientific and educational institutions in the world. Its interests include geography, archaeology and natural science, the promotion of environmental and historical...
. Many educational institutes and textbooks followed National Geographic's example in adopting the projection, and most of those still use it.