Map projections, explained

A globe cannot be flattened without tearing or stretching it. That is a theorem, not an opinion. Every world map is therefore a decision about what to keep true and what to let go. This page explains the three ways of deciding, why Mercator makes Greenland the size of Africa, and how the Equal Earth projection of 2018 keeps area exact while still looking like a map. The interactive version is at KUON EQUAL EARTH.

Why a flat map must lie somewhere

Carl Friedrich Gauss proved in 1827 that a curved surface such as a sphere cannot be mapped onto a plane while preserving all distances. This is the Theorema Egregium. Peel an orange and try to press the peel flat: it splits or it stretches. A map projection is a rule for how to stretch, and the rule decides what survives.

Three things compete. Angles, so that shapes are locally correct. Areas, so that the size of a region is correct. Distances or directions, at least from one point or along one line. No projection of the whole Earth keeps all three. Most keep one and sacrifice the others, or keep none exactly and spread the damage.

Three ways to choose

A conformal projection keeps angles. Small shapes are correct, meridians and parallels cross at right angles, and a compass bearing is a straight line. Mercator is the famous one. The price is area: it grows without limit toward the poles.

An equal-area projection keeps area. Any two regions on the map have the same ratio of areas as on the Earth. The price is shape: continents are sheared or squashed, more so toward the edges. Lambert's cylindrical projection of 1772, Mollweide of 1805, Gall-Peters and Equal Earth are all equal-area.

A compromise projection keeps neither exactly and tries to look right. Robinson of 1963 and Winkel Tripel of 1921 are the ones on most classroom walls. They are pleasant, and they are not equal-area: the Robinson map still draws Greenland about twice its true relative size.

There is a fourth kind worth naming: the azimuthal projections, which are correct in direction or distance from one centre point. They are the right tool for the poles, and that is why KUON POLAR uses a polar stereographic projection instead of Mercator.

What Mercator was built for

Gerardus Mercator published his world map in 1569 for navigation. On it, a line of constant compass bearing, a rhumb line, is straight. A navigator draws a straight line from port to port, reads the angle, and holds that course. No other projection does this, and for four centuries it was the reason Mercator existed.

The mathematics that makes bearings straight is the same mathematics that inflates area. At latitude φ, Mercator stretches distances by a factor of 1/cos φ in both directions, so area grows by 1/cos²φ. At the equator the factor is 1. At 30° it is 1.33. At 45° it is 2. At 60° it is 4. At 70° it is 8.5. At 80° it is 33. The poles cannot be drawn at all, which is why web maps stop at 85.05° and the world becomes a square.

Applied to countries, the effect is larger than most people expect. Integrating over the actual outlines, Mercator draws Japan at about 1.6 times its true relative area, Germany at 2.6, the United Kingdom at 2.9, Canada at 5.2, Norway at 9.4 and Greenland at 16.5. Africa, which straddles the equator, comes out at 1.1. That is how Greenland, at 2.15 million km², looks the size of Africa, at 30 million km². The true ratio is 1 to 14.

None of this makes Mercator wrong. It makes it a navigation chart. Every web map tile since 2005 uses Web Mercator, EPSG:3857, because a conformal projection keeps north up and shapes correct when you zoom into a city, which is what a street map is for. The problem begins only when the same projection is used to show the whole world at once and readers take the sizes literally.

Equal-area maps before Equal Earth

Johann Heinrich Lambert described the cylindrical equal-area projection in 1772. It is exact in area and brutal in shape: the tropics are stretched tall, the high latitudes are flattened. Karl Mollweide's ellipse of 1805 is equal-area and rounder, and it is still used for sky surveys and global climate maps.

James Gall proposed a variant of the cylindrical equal-area in 1855. In 1973 Arno Peters presented essentially the same projection as a corrective to the Eurocentric world map, and it entered politics as the Gall-Peters projection. The argument about relative size was sound; the map itself was widely disliked for its stretched continents, and in 1989 seven North American geographic organisations adopted a resolution urging publishers to stop using rectangular world maps altogether.

For the next three decades the practical answer was a compromise projection. Rand McNally commissioned Arthur H. Robinson's projection in 1963; the National Geographic Society used it from 1988 and switched to Winkel Tripel in 1998. These look natural. They are not equal-area, and the gap between an honest-looking map and an honest map stayed open.

Equal Earth: the formula and what it keeps

In 2018 Bojan Šavrič of Esri, Tom Patterson, then of the US National Park Service, and Bernhard Jenny of Monash University published the Equal Earth projection in the International Journal of Geographical Information Science (volume 33, number 3, pages 454 to 465, doi 10.1080/13658816.2018.1504949). Their stated aim was a projection with the visual appeal of Robinson that is strictly equal-area. They started from the Putniņš P4′ projection of 1934 and reshaped its outline with a polynomial.

The projection is pseudocylindrical: parallels are straight horizontal lines, the central meridian is straight, and the other meridians are curves. With longitude λ and latitude φ in radians, an auxiliary latitude θ is defined by sin θ = (√3 / 2) · sin φ. Then y = A₁θ + A₂θ³ + A₃θ⁷ + A₄θ⁹ and x = (2√3 / 3) · λ · cos θ / (A₁ + 3A₂θ² + 7A₃θ⁶ + 9A₄θ⁸), with A₁ = 1.340264, A₂ = −0.081106, A₃ = 0.000893, A₄ = 0.003796. The denominator of x is the derivative of y with respect to θ. That is the whole trick: wherever the parallels are squeezed together vertically, the meridians are spread apart horizontally by exactly the same factor, so area is preserved everywhere.

The inverse is not closed-form. Given y, solve the ninth-degree polynomial for θ by Newton's method (it converges in a few steps), then recover φ and λ. That is all an implementation needs, which is why PROJ added +proj=eqearth in version 5.2 within months of publication, and GDAL, QGIS, D3, Cartopy and ArcGIS followed.

Some properties follow from the coefficients. The poles are not points but lines, 59.2 percent as long as the equator. The whole map is 2.05 times as wide as it is tall. We verify the equal-area claim numerically on every build of this site: integrating the area scale over the globe on a 720 by 720 grid gives 12.56638058 against 4π = 12.56637061, a relative difference of 7.9 × 10⁻⁷, which is the discretisation error and nothing more. You can run the same integration in your browser on the map page with the Verify button, and run it for Mercator to watch it fail.

What Equal Earth gives up is shape at the edges: the high latitudes are compressed and the outer continents lean. Distances and bearings are not preserved. It is a map for seeing the world as a whole, not for steering a ship or surveying a field.

Tissot's indicatrix: seeing the distortion

Nicolas Auguste Tissot showed in 1859 that an infinitesimal circle on the sphere becomes an ellipse on any map. Draw those ellipses at regular intervals and the projection explains itself. On a conformal map all ellipses are circles of different sizes. On an equal-area map they all have the same area and different shapes. On a compromise map both vary. The map page draws them as circles of 2.5 degrees radius; switch projections and watch what changes.

Using Equal Earth in practice

Three coordinate reference systems are registered for it, differing only in the central meridian. EPSG:8857, WGS 84 / Equal Earth Greenwich, is centred on 0°. EPSG:8858, Equal Earth Americas, is centred on 90°W. EPSG:8859, Equal Earth Asia-Pacific, is centred on 150°E. Units are metres. The Japanese classroom convention of a Pacific-centred map corresponds to a central meridian near 135°E or 150°E, which the map page offers as a preset.

In QGIS (PROJ 5.2 or later): Project, Properties, CRS, type 8857 in the filter and select WGS 84 / Equal Earth Greenwich. Vector and raster layers reproject on the fly. For an Asia-centred map choose 8859.

With PROJ on the command line, the definition is +proj=eqearth +lon_0=0 +datum=WGS84 +units=m. With GDAL, reproject a raster with gdalwarp -t_srs EPSG:8857 in.tif out.tif and a vector file with ogr2ogr -t_srs EPSG:8857 out.gpkg in.gpkg.

In Python with pyproj: Transformer.from_crs("EPSG:4326", "EPSG:8857", always_xy=True).transform(lon, lat). With Cartopy, ax = plt.axes(projection=ccrs.EqualEarth()). In D3, d3.geoEqualEarth() is built in; this site uses its own implementation of the raw formula plugged into d3.geoProjection, so the code on the map page is the formula above.

One warning that applies to every equal-area map: do not measure distances on it. A kilometre grid drawn on Equal Earth is wrong everywhere except along the central meridian. For survey and positioning work use a conformal local system such as UTM or, in Japan, the plane rectangular coordinate system described in coordinates and height.

Which projection for which job

For a navigation chart or a street map that people zoom into, a conformal projection: Mercator, or Web Mercator for tiles. For a thematic map of the whole world, a choropleth, a density map, anything where the reader compares the size of regions, an equal-area projection, and Equal Earth is the one that reads as a map. For a wall map or a textbook, Equal Earth or a compromise projection, with the choice written on the map. For the Arctic and Antarctic, an azimuthal projection centred on the pole. For measuring, a local conformal system in metres.

The honest habit is to print the projection name on the map. A reader who knows which rule was applied can undo it in their head. A reader who does not is at the mercy of the rule.

The resolution of 4 September 2026

On 4 September 2026 the United Nations General Assembly adopted a resolution recommending the Equal Earth projection for world maps in place of Mercator. 164 member states voted in favour, including Japan. The United States voted against. Six abstained. The resolution was led by Togo and grew out of the Correct The Map campaign backed by the African Union, which argued that a century of Mercator classroom maps had shrunk Africa in the public mind. According to Reuters, the United States described the resolution as ideological and said it did not serve international peace.

A General Assembly resolution is a recommendation, not a law. What it changes is the default: from now on, a publisher who chooses Mercator for a world map is making a choice rather than following a habit. The numbers on this page are what that choice weighs.