Map projections

The Earth is round. Paper is flat.

To make a flat map, mapmakers use a plan called a projection. It is like a recipe for flattening the Earth. Different recipes give different maps.

Earth is a sphere, but paper and screens are flat. The way a mapmaker moves the round Earth's surface onto a flat map is called a map projection.

This cannot be done perfectly. Some places always get stretched or squashed. So mapmakers have invented many different projections, each one good for certain jobs. A sailor, a pilot and a scientist studying Antarctica might each choose a different one.

USGS defines a map projection as the method used to portray part or all of the spherical Earth on a flat surface, whether a paper map or a computer screen, and states that this cannot be done without some distortion. National Geographic describes the projection as the final perspective of a map, including which parts are distorted and which are represented more realistically, chosen by cartographers to suit the map's purpose.

Mapmakers and mathematicians have devised almost limitless ways to project the globe onto paper, and USGS scientists have designed projections for their own needs, such as the Space Oblique Mercator for mapping from satellites. Every projection has advantages and disadvantages. This topic explains how projections are grouped, what they can keep true, and how cartographers choose among them.

Mapmakers imagine three shapes touching the globe.

A flat sheet. A cone, like an ice-cream cone. A tube, like a paper towel roll. The map is drawn onto the shape, then laid flat.

Most projections belong to one of three families. In a planar projection, imagine a flat sheet touching the globe at one point; the map is most accurate at that point, and planar maps are often used for the North or South Pole.

In a conic projection, a cone sits over the globe like a hat, touching it along one or two lines of latitude. These maps are fairly accurate in the middle latitudes, so they are often used for the United States. In a cylindrical projection, a tube wraps around the globe, usually touching at the Equator. These maps are most accurate near the Equator and are often used for world maps.

National Geographic groups projections into three basic categories. Planar projections project the surface onto a plane touching the globe at a point; they are most accurate at their centers and often used for polar maps. Conical projections set a cone over the globe, tip above a pole, touching along one or two parallels; unrolled, the parallels become circular arcs and the meridians straight lines meeting at the pole, and the midlatitudes are represented fairly accurately, which is why such maps often show the United States. Cylindrical projections wrap a tube around the globe, usually touching along the Equator; regions near the Equator are most accurate in size and shape and those near the poles most distorted, and these are often used for world maps.

USGS uses the terms azimuthal (planar), conic and cylindrical, and adds a distinction: some projections are geometrically projected from a point at the globe's center, on its surface or at infinity, while others are defined mathematically. A cone can also be secant, cutting the globe along two standard parallels instead of touching along one.

A globe shows four things truly: which way, how far, what shape and how big.

A flat map can never show all four. Each projection picks what to keep.

A globe shows four things truly: directions, distances, shapes and areas (sizes). USGS explains that a flat map can show one or more of these, but never all four.

So projections have names for what they keep. A conformal projection keeps the shapes of small areas and the angles between directions, but sizes change. An equal-area projection keeps every area in the right proportion, but shapes get bent. An equidistant projection keeps distances true, but only from one point or along certain lines.

USGS frames every projection as a choice among four properties: true directions, true distances, true areas and true shapes. A flat map, or parts of one, can show one or more but never all.

Conformal maps keep angles and the shapes of small areas correct, but sizes of areas are distorted; Mercator, Transverse Mercator, Lambert Conformal Conic and Stereographic are conformal. Equal-area maps show all areas in true proportion to the same areas on Earth, but shapes are more or less distorted on every equal-area map; examples include Albers Equal Area Conic, Lambert Azimuthal Equal Area and the Sinusoidal. On an equidistant map, distances are true only along particular lines, such as those radiating from the projection's center; the Azimuthal Equidistant projection gives true distances and directions from its center point to every other place. Some projections keep none of these perfectly and instead balance the errors, as the next sections show.

The Mercator map was made in 1569.

Sailors liked it, because a straight line on it shows which way to steer. But it makes lands near the poles look much too big.

One cylindrical projection was introduced by Gerardus Mercator in 1569. On a Mercator map, any straight line keeps the same compass direction the whole way, so sailors could draw a straight course and steer by it. That made it valuable for navigation.

But the lines of latitude get farther and farther apart toward the poles, so places far from the Equator look much too big. Greenland looks bigger than South America. Mercator maps cannot even show the poles. Other cylindrical maps, like the Miller projection of 1942, look similar but stretch the far north and south less.

Mercator's projection (1569) is conformal, and any straight line on it is a rhumb line, a line of constant compass direction; USGS notes a rhumb line is usually not the shortest route, so some navigators paired it with the Gnomonic, on which every straight line is a great circle. Its parallels are spaced increasingly far apart toward the poles, which are not shown at all. Distances are true only along the Equator and reasonably correct within 15° of it, while areas and shapes of large regions are distorted, extremely so near the poles: Britannica cites Greenland appearing larger than South America.

Turned on its side, the same idea becomes the Transverse Mercator, accurate within about 15° of a chosen central meridian; USGS uses it for many quadrangle maps at scales from 1:24,000 to 1:250,000, and its 1:24,000 maps are nearly correct in every respect. The Miller Cylindrical (O. M. Miller, 1942) looks like Mercator and avoids some of its exaggeration, but it is not useful for navigation and shows neither shapes nor areas without distortion. USGS designed the Space Oblique Mercator for Landsat satellite images, with no distortion along the satellite's ground track.

Some maps are made for one country. Others are made for the North or South Pole.

Mapmakers choose a shape that fits the place. Then that place looks just right.

Conic projections suit places that stretch from east to west in the middle latitudes, like the United States. The Albers projection, from 1805, keeps areas true, and USGS uses it for maps of the 48 connected states. The Lambert Conformal Conic, from 1772, keeps shapes true and is one of the most widely used projections in the United States.

Planar projections, also called azimuthal, are centered on one point. The Stereographic projection is used for maps of Antarctica. The Azimuthal Equidistant shows true distances from its center, and its polar version is used in the United Nations emblem.

The Albers Equal Area Conic (presented by H. C. Albers in 1805) is equal-area, with true scale along two standard parallels; USGS uses it for maps of the conterminous United States, where its maximum scale error is about 1¼ percent. The Lambert Conformal Conic (Lambert, 1772) keeps shapes and is used for many USGS topographic maps and the State Base Map series; USGS calls it one of the most widely used projections in the United States.

Azimuthal projections radiate from a center point. The Stereographic, conformal and ascribed to Hipparchus in the 2nd century BCE, is used by USGS for maps of Antarctica and for navigation charts above 80° latitude. The Gnomonic, considered the oldest projection and ascribed to Thales in the 6th century BCE, puts every straight line on a great circle, so some navigators use it with the Mercator to find the shortest path and seismologists use it because seismic waves tend to travel along great circles. The Azimuthal Equidistant gives true distances from its center, useful for airline distances and radio work; its polar aspect appears in the United Nations emblem. The Lambert Azimuthal Equal Area suits regions that extend equally in all directions, such as Asia and the Pacific.

There is no perfect flat map.

Some maps try to make the whole world look about right. Mapmakers choose the map that fits the job best.

Some world maps do not keep any one thing perfectly true. Instead, they balance the errors so the whole world looks about right. The Robinson projection, made by Arthur H. Robinson in 1963, does this. It shows far-northern lands such as Canada and Greenland closer to their true sizes than Mercator does, though Greenland is a bit squashed.

The U.S. Geological Survey's advice is simple: there is no best projection. A mapmaker picks the one that suits the job, keeping the most important things as true as possible.

A compromise projection keeps none of the four properties exactly; instead every point has some distortion. Arthur H. Robinson's projection (1963) uses tables of coordinates rather than formulas to make the world "look right": USGS says it balances the size and shape of high-latitude lands better than Mercator, Van der Grinten or Mollweide, showing the Soviet Union, Canada and Greenland truer to size, though Greenland is compressed. Distortion is very low along the Equator and within 45° of the center and greatest near the poles. USGS's 1993 poster noted that it had been adopted for National Geographic's world maps in 1988 and might replace Mercator in many classrooms.

USGS's general rule follows: there is no "best" projection; the mapmaker must select the one best suited to the needs, reducing distortion of the most important features. Scale matters as much as projection: the extreme high-latitude distortion of a small-scale Mercator world map almost disappears on a large-scale Transverse Mercator map of a small area. Reading any map well includes asking which projection it uses and what that projection sacrifices.