# Winkel tripel projection

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The Winkel tripel projection (Winkel III), a modified azimuthal map projection of the world, 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 of the equirectangular projection and the Aitoff projection: [1] The name tripel (German for "triple") refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance. [2]

## Algorithm

{\displaystyle {\begin{aligned}x&={\frac {1}{2}}\left(\lambda \cos \varphi _{1}+{\frac {2\cos \varphi \sin {\frac {\lambda }{2}}}{\operatorname {sinc} \alpha }}\right),\\y&={\frac {1}{2}}\left(\varphi +{\frac {\sin \varphi }{\operatorname {sinc} \alpha }}\right),\end{aligned}}}

where λ is the longitude relative to the central meridian of the projection, φ is the latitude, φ1 is the standard parallel for the equirectangular projection, sinc is the unnormalized cardinal sine function (with the discontinuity removed), and

${\displaystyle \alpha =\arccos \left(\cos \varphi \cos {\frac {\lambda }{2}}\right).}$

In his proposal, Winkel set

${\displaystyle \varphi _{1}=\arccos {\frac {2}{\pi }}.}$

A closed-form inverse mapping does not exist, and computing the inverse numerically is somewhat complicated. [3]

## Comparison with other projections

David M. Goldberg and J. Richard Gott III 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 ellipticity and area errors, and the smallest skewness of any of the projections they studied. [4] 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 projections. [5]

In 1998, the Winkel tripel projection replaced the Robinson projection as the standard projection for world maps made by the National Geographic Society. [2] Many educational institutes and textbooks followed National Geographic's example in adopting the projection, and most of those still use it. [6] [7]

## Related Research Articles

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred in marine navigation because ships can sail in a constant compass direction for long stretches, reducing the difficult, error-prone course corrections that otherwise would be needed frequently when sailing other kinds of courses. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator but accelerates with increasing latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north.

The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for global maps of the world or night sky. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.

The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

The Bottomley map projection is an equal area map projection defined as:

The Universal Transverse Mercator (UTM) is a system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.

The Aitoff projection is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection, Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes. Expressed simply:

The Hammer projection is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2:1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.

The Cassini projection is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:

The Kavrayskiy VII projection is a map projection invented by Soviet cartographer Vladimir V. Kavrayskiy in 1939 for use as a general-purpose pseudocylindrical projection. Like the Robinson projection, it is a compromise intended to produce good-quality maps with low distortion overall. It scores well in that respect compared to other popular projections, such as the Winkel tripel, despite straight, evenly spaced parallels and a simple formulation. Regardless, it has not been widely used outside the former Soviet Union.

Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a). They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such as great-circle distance.

The equidistant conic projection is a conic map projection commonly used for maps of small countries as well as for larger regions such as the continental United States that are elongated east-to-west.

The Nicolosi globular projection is a map projection invented about the year 1,000 by the Iranian polymath al-Biruni. As a circular representation of a hemisphere, it is called globular because it evokes a globe. It can only display one hemisphere at a time and so normally appears as a "double hemispheric" presentation in world maps. The projection came into use in the Western world starting in 1660, reaching its most common use in the 19th century. As a "compromise" projection, it preserves no particular properties, instead giving a balance of distortions.

The Strebe 1995 projection, Strebe projection, Strebe lenticular equal-area projection, or Strebe equal-area polyconic projection is an equal-area map projection presented by Daniel "daan" Strebe in 1994. Strebe designed the projection to keep all areas proportionally correct in size; to push as much of the inevitable distortion as feasible away from the continental masses and into the Pacific Ocean; to keep a familiar equatorial orientation; and to do all this without slicing up the map.

The Eckert-Greifendorff projection is an equal-area map projection described by Max Eckert-Greifendorff in 1935. Unlike his previous six projections, It is not pseudocylindrical.

## References

1. Snyder, John P. (1993). Flattening the Earth: Two Thousand Years of Map Projections. Chicago: University of Chicago Press. pp. 231–232. ISBN   0-226-76747-7 . Retrieved 2011-11-14.
2. "Winkel Tripel Projections". Winkel.org. Retrieved 2011-11-14.
3. Ipbüker, Cengizhan; Bildirici, I.Öztug (2002). "A General Algorithm for the Inverse Transformation of Map Projections Using Jacobian Matrices" (PDF). Proceedings of the Third International Symposium Mathematical & Computational Applications. Third International Symposium Mathematical & Computational Applications September 4–6, 2002. Konya, Turkey. Selcuk, Turkey. pp. 175–182. Archived from the original (PDF) on 20 October 2014.
4. Goldberg, David M.; Gott III, J. Richard (2007). "Flexion and Skewness in Map Projections of the Earth" (PDF). Cartographica. 42 (4): 297–318. arXiv:. doi:10.3138/carto.42.4.297 . Retrieved 2011-11-14.
5. Capek, Richard (2001). "Which is the best projection for the world map?" (PDF). Proceedings of the 20th International Cartographic Conference. Beijing, China. 5: 3084–93. Retrieved 2018-11-15.
6. "NG Maps Print Collection – World Political Map (Bright Colored)". National Geographic Society. Retrieved 1 October 2013. This latest world map ... features the Winkel Tripel projection to reduce the distortion of land masses as they near the poles.
7. "Selecting a Map Projection – National Geographic Education". National Geographic Society. Retrieved 1 October 2013.