# Cylindrical equal-area projection

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In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.

## Cylindrical projections

The term "normal cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude are mapped to horizontal lines (or, mutatis mutandis, more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines).

The mapping of meridians to vertical lines can be visualized by imagining a cylinder (of which the axis coincides with the Earth's axis of rotation) wrapped around the Earth and then projecting onto the cylinder, and subsequently unfolding the cylinder.

By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude is given by φ):

The only cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching (cos  φ): equal-area cylindrical (with many named specializations such as Gall–Peters or Gall orthographic, Behrmann, and Lambert cylindrical equal-area). This divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes.

Any particular cylindrical equal-area map has a pair of identical latitudes of opposite sign (or else the equator) at which the east–west scale matches the north–south scale.

## Description

### Formulae

All cylindrical equal-area projections use the formula:

{\displaystyle {\begin{aligned}x&=(\lambda -\lambda _{0})\cos \varphi _{0}\\y&={\frac {\sin \varphi }{\cos \varphi _{0}}}\end{aligned}}}

where λ is the longitude, λ0 is the central meridian, φ is the latitude, and φ0 is the standard latitude, [1] all expressed in radians.

Some cartographers prefer to work in degrees, rather than radians, and use the equivalent formula:

{\displaystyle {\begin{aligned}x&={\frac {\pi (\lambda -\lambda _{0})\cos \varphi _{0}}{180^{\circ }}}\\y&={\frac {\sin \varphi }{\cos \varphi _{0}}}\end{aligned}}}

### Simplified formula

Stripping out unit conversion and uniform scaling, the formulae may be written:

{\displaystyle {\begin{aligned}x&=(\lambda -\lambda _{0})S\\y&=\sin \varphi \end{aligned}}}

Hence the sphere is mapped onto a stretched vertical cylinder. The stretch factor S is what distinguishes the variations of cylindric equal-area projection.

### Discussion

The various specializations of the cylindric equal-area projection differ only in the ratio of the vertical to horizontal axis. This ratio determines the standard parallel of the projection, which is the parallel at which there is no distortion and along which distances match the stated scale. There are always two standard parallels on the cylindric equal-area projection, each at the same distance north and south of the equator. The standard parallels of the Gall–Peters are 45° N and 45° S. Several other specializations of the equal-area cylindric have been described, promoted, or otherwise named. [2] [3] [4] [5] [6]

Named specializations of the cylindrical equal-area projection
ProjectionImageCreator (year)Standard parallels north and southWidth-to-height aspect ratio
cylindrical equal-area
(base projection for all the others)
φ0π(cos φ0)2
Lambert Johann Heinrich Lambert (1772)Equator (0°)π ≈ 3.142
Behrmann Walter Behrmann (1910)30°3π/4 ≈ 2.356
Smyth equal-surface
= Craster rectangular
Charles Piazzi Smyth (1870)${\displaystyle \arccos {\sqrt {\tfrac {2}{\pi }}}}$ ≈ 37°04′17″2
Trystan Edwards Trystan Edwards (1953)37°24′≈ 1.983
Hobo–Dyer Mick Dyer (2002)37°30′≈ 1.977
Gall–Peters
= Gall orthographic
= Peters
James Gall (1855)
Promoted by Arno Peters as his own invention(1967)
45°π/2 ≈ 1.571
BalthasartM. Balthasart (1935)50°≈ 1.298
Tobler's world in a square Waldo Tobler (1986)${\displaystyle \arccos {\sqrt {\tfrac {1}{\pi }}}}$ ≈ 55°39′14″1

## History

The invention of the Lambert cylindrical equal-area projection is attributed to the Swiss mathematician Johann Heinrich Lambert in 1772. [7] Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios. See Named specializations table.

The Tobler hyperelliptical projection, first described by Tobler in 1973, is a further generalization of the cylindrical equal-area family.

The HEALPix projection is an equal-area hybrid combination of: the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere; and an interrupted Collignon projection, for the polar regions.

## Related Research Articles

The Gall–Peters projection is a rectangular map projection that maps all areas such that they have the correct sizes relative to each other. Like any equal-area projection, it achieves this goal by distorting most shapes. The projection is a particular example of the cylindrical equal-area projection with latitudes 45° north and south as the regions on the map that have no distortion.

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 by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. 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 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 transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

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 Craig retroazimuthal map projection was created by James Ireland Craig in 1909. It is a modified cylindrical projection. As a retroazimuthal projection, it preserves directions from everywhere to one location of interest that is configured during construction of the projection. The projection is sometimes known as the Mecca projection because Craig, who had worked in Egypt as a cartographer, created it to help Muslims find their qibla. In such maps, Mecca is the configurable location of interest.

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 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 sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.

The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvano and Honter's usages were approximate, however, and it is not clear they intended to be the same projection.

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

A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten.

The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. Although scale and shape are not preserved, distortion is minimal between the standard parallels.

In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points.

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 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 Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III.

The rectangular polyconic projection is a map projection was first mentioned in 1853 by the U.S. Coast Survey, where it was developed and used for portions of the U.S. exceeding about one square degree. It belongs to the polyconic projection class, which consists of map projections whose parallels are non-concentric circular arcs except for the equator, which is straight. Sometimes the rectangular polyconic is called the War Office projection due to its use by the British War Office for topographic maps. It is not used much these days, with practically all military grid systems having moved onto conformal projection systems, typically modeled on the transverse Mercator projection.

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.

## References

1. Map Projections – A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp.76–85
2. Snyder, John P. (1989). An Album of Map Projections p. 19. Washington, D.C.: U.S. Geological Survey Professional Paper 1453. (Mathematical properties of the Gall–Peters and related projections.)
3. Monmonier, Mark (2004). Rhumb Lines and Map Wars: A Social History of the Mercator Projection p. 152. Chicago: The University of Chicago Press. (Thorough treatment of the social history of the Mercator projection and Gall–Peters projections.)
4. Smyth, C. Piazzi. (1870). On an Equal-Surface Projection and its Anthropological Applications. Edinburgh: Edmonton & Douglas. (Monograph describing an equal-area cylindric projection and its virtues, specifically disparaging Mercator's projection.)
5. Weisstein, Eric W. "Cylindrical Equal-Area Projection." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CylindricalEqual-AreaProjection.html
6. Tobler, Waldo and Chen, Zi-tan(1986). A Quadtree for Global Information Storage. http://www.geog.ucsb.edu/~kclarke/Geography232/Tobler1986.pdf
7. Mulcahy, Karen. "Cylindrical Projections". City University of New York . Retrieved 2007-03-30.