Mapping the World

Young Astronomers Blog, Volume 29, Number 4.

The world is round (actually it is an ellipsoid, but close enough).  Maps are flat. This difference doesn’t seem like much. But it is!

Because the Earth is round, its surface has what is known as a spherical geometry. Here the sum of the angles in a triangle add up to more than 180^o and there are no parallel lines. It is distinctly different from the flat “Euclidian” geometry we might be familiar with from high school.

The Earth has a finite closed surface, yet it is unbounded. Lines are described by “geodesics” (great circles that cut the Earth in half). These “lines” define the shortest path between any two points. Lines of longitude are geodesics and run north and south meeting at the poles. An infinite number run through a single point (at the poles); therefore, they are not parallel to each other. “Lines” of latitude run east and west, appear to be parallel, but they are not geodesics.

In the Northern Hemisphere airplanes follow a path that takes them toward the north (or south in the Southern Hemisphere). They are simply following the geodesics that represent the shortest path on the spherical Earth.

Globes are spherical and provide accurate maps of the Earth (and other planets). However, it is difficult to stuff a globe into your glove compartment or bring one up on your phone or GPS device. So, there is a need for flat two-dimensional “Euclidian” maps. The challenge is how to project a spherical surface on a flat map. It isn’t easy.

One of the first to tackle this problem was Gerardus Mercator, who developed one of the most popular maps in 1569. He wanted to create a map that aided in the navigation of ships. The map was rectangular with “lines” (latitude and longitude) that were parallel in the Euclidian sense. The idea was that, to navigate from one place to another, just draw a line, find the direction with a compass, and go.

The Mercator projection was used for many years and became something of a standard, but it created a “European bias” in the process. It made northern areas appear larger and equatorial areas smaller. Greenland, for example, looks huge, although it is not. Africa appears the same size as North America and Greenland, although it is much larger.

At a press conference in 1974, Arno Peters stood up and said No! This needed to change. Peters introduced a map that shows everything with the correct area. Africa is larger than North America, and Greenland is scrunch up near the pole. However, it tends to stretch things out giving everything a long and stringy look. It also turns out that this same concept was published by James Gall in 1885, so it is often referred to as the Gall-Peters projection.

If you’ve watched the TV program The West Wing, you might remember the episode in which the staff has a discussion with the “Organization of Cartographers for Social Equality” about the Mercator and Peters projections.

Others were looking at the problems with the Mercator projection long before Peters. They decided to move away from rectangular maps to more circular and “elliptical” maps.

In 1893, Alphons J. van der Grinten created a map using a circle rather than a rectangle. He offered three other versions of his van der Grinten projection in 1904. These projections had curved “lines”, but still enlarged the land areas to the north and south. It was, however, adopted by the National Geographic Society in 1922.

In 1805, Karl Brandan Mollweide developed an “elliptical” shaped map. It was reintroduced in 1857 by Jacques Babinet. The Mollweide/Babinet projection correctly represents the area of the continents, although distorts their shape near the poles. The Mollweide projection is often used for all-sky astronomy maps such as those of the Cosmic Microwave Background (CMB).

In 1963, Arthur H. Robinson developed a map somewhat similar to the Mollweide projection, but with a flattening at the poles. He published the details in 1974. The Robinson projection had a similar “elliptical” shape, but still slightly distorted some of the land masses near the poles. It was close enough, and the National Geographic Society adopted it in 1988.

Oswald Winkel proposed a more circular “elliptical” map in 1921. The Winkel Tripel projection was a balance of three characteristics, and, as such, includes the German word Tripel in its name. It didn’t get much attention early on. However, the National Geographic Society adopted this projection in 1998 renewing interest in Winkel’s approach.

All these projections have distortions of one type or another. The idea is to minimize a particular distortion by achieving one of the four criteria below. Some, including Robinson and Winkel Tripel, compromise among the criteria to achieve the best results.

• Conformal preserves the angles and shapes (Mercator).
• Equal Area preserves the correct areas (Mollweide).
• Equidistant preserves the distance between locations.
• True Direction preserves the direction from a location.

In 1859, French mathematician Nicolas Auguste Tissot develop a methodology (Tissot’s Indicatrix) to identify the inaccuracies in a mapping projection. Tissot’s approach was to place ellipses (indicatrices) over the map at specific intervals. A globe will show perfectly spaced and equally sized circles. However, for flat projections, the shape and spread of the ellipses are in proportion to the distortions. Wikipedia shows the Tissot’s indicatrices for several of the maps discussed above (Mercator, Gall-Peters, Mollweide, Robinson, and Winkel Tripel).

For an in-depth description of many of these mapping projections, including the Tissot Indicatrix for each, see the USGS’s An album of map projections.

In 2007, David Goldberg and Richard Gott took this a step further and introduced the “Goldberg-Goth” index, which measures the sum of the squares of six individual distortion indices, with 0 being perfect. The Mercator projection received 8.308, while the Winkel Tripel projection was one of the best at 4.563.

Just recently, Goldberg and Gott, along with Robert Vanderbei, proposed a new and somewhat innovative way to improve map projections – make them two sided. In doing so, their distortion index dropped to 0.881. Their paper discusses the application of their index and has several examples of global maps. It includes a few maps of solar system objects and a video showing a two-sided world map.

With all the innovations and progress in mapping projections, we still need the maps on our smart phones. They use … the Web Mercator projection. Oh no, we’re back to the beginning. The Mercator projection, however, has several advantages that are useful for online maps. Remember it was designed to help with navigation (think GPS). It also doesn’t distort while zooming in and out. So, Mercator works in this situation.

Today, maps are used for many different applications, and with this there are several different projection techniques applicable. Most fall into one of three categories based on the two-dimensional form the map is projected on to (cylinder, cone, or plane). The rectangular maps discussed above fall into the class of Cylindrical projections. The first two “elliptical” maps are considered Pseudo-cylindrical projections. There are two other common techniques (Conical and Planal/Azimuthal projections), although neither is useful for mapping the entire globe. The Winkel Tripel is, however, considered a Modified Azimuthal projection.

Therefore, if you thought a map was just a map, it isn’t that simple. If you need a map of the world, there are many choices, each with its advantages and disadvantages. You could study cartography or invest in a geographic information system (GIS). For me, the answer is – find a globe. 😊