Shortest distance

In January 2011, Lufthansa reported that one of its Boeing 747 aircraft had aborted its flight to San Francisco over Greenland due to oil loss in one of the four engines and returned to Frankfurt am Main.

Looking at a map of the world, one wonders in amazement what the Boeing 747 was “doing” on a direct flight Frankfurt am Main — San Francisco over Greenland. If, on the other hand, one looks at a globe, it immediately becomes clear that the shortest route from Frankfurt am Main to San Francisco runs precisely over Greenland (cf. Figure 1) and not — as a world map (cf. Figure 2) would suggest — at a distance of about 100 miles north “past” New York.

An experiment in MATHEMATICS ADVENTURE LAND makes this clear by comparing distances on the globe and on a world map, as the following figures show:

Figure 1: Route from Frankfurt am Main to San Francisco on the globe
Figure 2: Route from Frankfurt am Main to San Francisco on the map

Of course, the shortest connection between two points in a plane is a straight line. Its length is the so-called Euclidean distance (Figure 2 shows this clearly).

BUT:

The role of straight lines in spherical geometry (that is, geometry on the sphere) belongs to the so-called great circles. Great circles are circles on the sphere whose (so-called “Euclidean”) center is the center of the sphere. Examples of great circles on the globe are the equator and the meridians. A great circle is obtained by intersecting the surface of the sphere with a plane containing the center of the sphere.

That is, the shortest connection between two points on a sphere (that is, in particular, between two cities on the globe) is part of a great circle. In the concrete case of the shortest connection from Frankfurt am Main to San Francisco, the corresponding great circle — i.e. the shortest connection between these two cities — leads right across Greenland.

A (plane) world map does not reflect this fact, because the mapping of a sphere onto a plane necessarily requires that at least one of the following properties must be renounced in this mapping: namely the angular fidelity or the surface fidelity or the distance fidelity or the direction fidelity.


And now … the mathematics of it:

In MATHEMATICS ADVENTURE LAND a world map is shown next to a globe, which was created by a — in principle — cylindrical projection. A (simple) cylindrical projection is an image where the surface of a sphere is projected onto a cylinder. Specifically, in a so-called tangential cylindrical projection, the sphere touches the cylinder at one of its great circles (e.g., at the equator). One can imagine this projection as follows: The light rays emanating from a light source at the center of the sphere (assumed to be translucent) then map the surface of the sphere onto the inside of the cylinder (cf. Figure 3). This is therefore a central projection.

Figure 3: Cylindrical projection

This projection — it is true to the angle — is often (but probably wrongly!) called the Mercator projection. Although the actual Mercator projection is a cylindrical projection, there is an essential difference to the projection method shown in figure 3. In the actual Mercator projection, the image created by (central) projection is distorted in the direction of the cylinder axis in such a way that the scale in the vertical north-south direction is the same at every point as in the horizontal east-west direction. Consequently, the scale changes continuously from the equator in the direction of the north pole on the one hand and in the direction of the south pole on the other hand, but it is the same at every place on the globe in vertical and horizontal direction.

The Mercator projection is therefore not an optically (for example by the course of light rays) describable projection, but only analytically, i.e. by mathematical illustration properties, to produce. One uses, in order to be able to imagine the process of this projection, the following plausibility consideration: One imagines the earth as a spherical balloon and brings this into a glass cylinder, so that the equator (on the balloon) touches exactly the wall of the enveloping cylinder. If the balloon is inflated, the areas south and north of the equator are increasingly pressed against the cylinder wall in the respective polar direction. The scale thus changes steadily and in both directions. The construction of a Mercator map is shown in the following figure 4:

Figure 4: Construction of the Mercator map

For clarification, two circles are drawn on a sphere. The first circle at 0°, i.e. at the equator, the second circle at 60°. If you now unroll the surface of the sphere, you get two curvilinear bounded triangles, which touch each other at the equator and continuously move away from each other in both polar directions. If we now stretch the surfaces in such a way that there are no more gaps, we see that the upper circle has been distorted. Due to the stretching in x-direction it became an ellipse. In the third part of the Mercator construction, the map is now stretched so that all circles become circles again. Thus, the distances between the latitudes increase towards the North Pole as well as towards the South Pole. This distortion also leads to the fact that Mercator maps usually only reach 60° or 70° latitude in northern or southern direction. Therefore, the representation of the poles must be omitted.

The Mercator design is angle-faithful. Therefore, so-called rhumb lines (i.e. curves which always intersect the meridians at the same angle) become curves which consist piecewise of straight lines. But: The Mercator design is not true to the area, as figure 2 shows — even if only partially: Although the island of Greenland has only about 1/15 of the area of the African continent, it appears on a world map about as large as Africa:

Figure 5: World map in Mercator projection

Literature

[1] Kuntz, E.: Kartennetzentwurfslehre: Grundlage und Anwendungen, 2. Auflage, Karlsruhe, 1990.

[2] Schröder, E.: Kartenentwürfe der Erde, Frankfurt am Main, 1988.