# Mollweide projection

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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.

## Contents

The projection was first published by mathematician and astronomer Karl (or Carl) Brandan Mollweide (1774–1825) of Leipzig in 1805. It was reinvented and popularized in 1857 by Jacques Babinet, who gave it the name homalographic projection. The variation homolographic arose from frequent nineteenth-century usage in star atlases. [1]

## Properties

The Mollweide is a pseudocylindrical projection in which the equator is represented as a straight horizontal line perpendicular to a central meridian one-half its length. The other parallels compress near the poles, while the other meridians are equally spaced at the equator. The meridians at 90 degrees east and west form a perfect circle, and the whole earth is depicted in a proportional 2:1 ellipse. The proportion of the area of the ellipse between any given parallel and the equator is the same as the proportion of the area on the globe between that parallel and the equator, but at the expense of shape distortion, which is significant at the perimeter of the ellipse, although not as severe as in the sinusoidal projection.

Shape distortion may be diminished by using an interrupted version. A sinusoidal interrupted Mollweide projection discards the central meridian in favor of alternating half-meridians which terminate at right angles to the equator. This has the effect of dividing the globe into lobes. In contrast, a parallel interrupted Mollweide projection uses multiple disjoint central meridians, giving the effect of multiple ellipses joined at the equator. More rarely, the projection can be drawn obliquely to shift the areas of distortion to the oceans, allowing the continents to remain truer to form.

The Mollweide, or its properties, has inspired the creation of several other projections, including the Goode's homolosine, van der Grinten and the Boggs eumorphic. [4]

## Mathematical formulation

The projection transforms from latitude and longitude to map coordinates x and y via the following equations: [5]

{\displaystyle {\begin{aligned}x&=R{\frac {2{\sqrt {2}}}{\pi }}\left(\lambda -\lambda _{0}\right)\cos \theta ,\\[5px]y&=R{\sqrt {2}}\sin \theta ,\end{aligned}}}

where θ is an auxiliary angle defined by

${\displaystyle 2\theta +\sin 2\theta =\pi \sin \varphi \qquad (1)}$

and λ is the longitude, λ0 is the central meridian, φ is the latitude, and R is the radius of the globe to be projected. The map has area 4πR2, conforming to the surface area of the generating globe. The x-coordinate has a range of [−2R2, 2R2], and the y-coordinate has a range of [−R2, R2].

Equation (1) may be solved with rapid convergence (but slow near the poles) using Newton–Raphson iteration: [5]

{\displaystyle {\begin{aligned}\theta _{0}&=\varphi ,\\\theta _{n+1}&=\theta _{n}-{\frac {2\theta _{n}+\sin 2\theta _{n}-\pi \sin \varphi }{2+2\cos 2\theta _{n}}}.\end{aligned}}}

If φ = ±π/2, then also θ = ±π/2. In that case the iteration should be bypassed; otherwise, division by zero may result.

There exists a closed-form inverse transformation: [5]

{\displaystyle {\begin{aligned}\varphi &=\arcsin {\frac {2\theta +\sin 2\theta }{\pi }},\\[5px]\lambda &=\lambda _{0}+{\frac {\pi x}{2R{\sqrt {2}}\cos \theta }},\end{aligned}}}

where θ can be found by the relation

${\displaystyle \theta =\arcsin {\frac {y}{R{\sqrt {2}}}}.\,}$

The inverse transformations allow one to find the latitude and longitude corresponding to the map coordinates x and y.

## Related Research Articles

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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:

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The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert II is the Eckert I projection.

The Boggs eumorphic projection is a pseudocylindrical, equal-area map projection used for world maps. Normally it is presented with multiple interruptions. Its equal-area property makes it useful for presenting spatial distribution of phenomena. The projection was developed in 1929 by Samuel Whittemore Boggs (1889–1954) to provide an alternative to the Mercator projection for portraying global areal relationships. Boggs was geographer for the United States Department of State from 1924 until his death. The Boggs eumorphic projection has been used occasionally in textbooks and atlases.

The Equal Earth map projection is an equal-area pseudocylindrical projection for world maps, invented by Bojan Šavrič, Bernhard Jenny, and Tom Patterson in 2018. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the relative size of areas. The projection equations are simple to implement and fast to evaluate.

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. Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 112113, ISBN   0-226-76747-7.
2. Gannon, Megan (December 21, 2012). "New 'Baby Picture' of Universe Unveiled". Space.com . Retrieved December 21, 2012.
3. Bennett, C.L.; Larson, L.; Weiland, J.L.; Jarosk, N.; Hinshaw, N.; Odegard, N.; Smith, K.M.; Hill, R.S.; Gold, B.; Halpern, M.; Komatsu, E.; Nolta, M.R.; Page, L.; Spergel, D.N.; Wollack, E.; Dunkley, J.; Kogut, A.; Limon, M.; Meyer, S.S.; Tucker, G.S.; Wright, E.L. (2013). "Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results". The Astrophysical Journal Supplement Series . 208 (2): 20. arXiv:. Bibcode:2013ApJS..208...20B. doi:10.1088/0067-0049/208/2/20.
4. Map Projections – A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 249252
5. Weisstein, Eric W. "Mollweide Projection". MathWorld .