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.^{ [1] }

The features of the Equal Earth projection include:^{ [2] }^{ [3] }

- The curved sides of the projection suggest the spherical form of Earth.
- Straight parallels make it easy to compare how far north or south places are from the equator.
- Meridians are evenly spaced along any line of latitude.
- Software for implementing the projection is easy to write and executes efficiently.

According to the creators, the projection was created in response to the decision of the Boston public schools to adopt the Gall-Peters projection for world maps in March 2017, which generated a controversy in the world of cartography. At that time Šavrič, Patterson and Jenny sought alternative map projections of equal areas for world maps, but could not find any that met their aesthetic criteria. Therefore, they created a new projection that had more visual appeal compared to existing projections of equal areas.^{ [3] }^{ [4] }^{ [5] }^{ [6] } They claim that they need a world map that shows continents and countries in real size with each other.^{ [2] }

The projection is formulated as the equations

where

and refers to latitude and to longitude.

The first known thematic map published using the Equal Earth projection is a map of the global mean temperature anomaly for July 2018, produced by the NASA’s Goddard Institute for Space Studies.^{ [7] }

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

The **haversine formula** determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the **law of haversines**, that relates the sides and angles of spherical triangles.

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 **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 **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:

This is a **table of orthonormalized spherical harmonics** that employ the Condon-Shortley phase up to degree * = 10. Some of these formulas give the "Cartesian" version. This assumes **x*, *y*, *z*, and *r* are related to and through the usual spherical-to-Cartesian coordinate transformation:

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.

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.

The **van der Grinten projection** is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion.

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.

In physics and mathematics, the **solid harmonics** are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the *regular solid harmonics*, which vanish at the origin and the *irregular solid harmonics*, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

In mathematics, **vector spherical harmonics** (**VSH**) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

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 **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 **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.

- ↑ Šavrič, Bojan; Patterson, Tom; Jenny, Bernhard (2018-08-07). "The Equal Earth map projection".
*International Journal of Geographical Information Science*.**33**(3): 454–465. doi:10.1080/13658816.2018.1504949. - 1 2 "Equal Earth projection".
*shadedrelief.com*. Retrieved 2018-08-23. - 1 2 Morales, Aurelio. "La nueva proyección Equal Earth: todo lo que debes saber" (in Spanish). Valladolid: MappingGIS. Retrieved January 24, 2020.
- ↑ "Equal Earth: un mapamundi más preciso que muestra el tamaño real de África" (in Spanish). N+1. August 22, 2018. Retrieved January 24, 2020.
- ↑ "Equal Earth: Idean un nuevo mapa del mundo basado en un mapa del 1569" (in Spanish). Código Oculto. Retrieved January 24, 2020.
- ↑ "Colección cartográfica - La proyección Equal-Earth".
*visionscarto*(in Spanish). Retrieved January 24, 2020. - ↑ "NASA GISS on Twitter".
*Twitter*. Retrieved 2018-08-23.

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