Do you think that’s air you’re breathing?

Morpheus

Navigational charts are an integral part of sailors’ lives. Today, maps are read in electronic format, or via navigation applications, but their construction remains substantially the same as in the early centuries: a rectangle with symbols representing the world. They are essential tools for passage planning and for navigation.

We are all familiar, to varying degrees, with the components of a navigational chart: symbolism, longitude and latitude coordinates, and the fact that it is usually rectangular. Some may also be aware of the role of the character in the image at the beginning of this text (Mercator). In short, maps are useful and familiar, so much that we do not even think about them anymore.

However, a little enquiry as to how they display the world makes one quickly realize that they distort reality. I am not referring to some shady conspiracy theory, nor to the obvious fact that everything is on a smaller scale, but about a result of geometry established ever since Carl Fredrich Gauss became interested in the curvature of the earth (in 1837): any map is necessarily a distortion of reality.

There is nothing better than a few examples to get a hold of the distortions. *Jean-du-Sud* made an Atlantic crossing in 1988. Her departure point was south of Newfoundland and her arrival was at the western tip of Ireland (image below).

Here is my first question: did *Jean-du-Sud* take the shortest route? For those who have not yet grasped the subtlety of what the map shows, I will take it a step further and present a second image: it shows the same route (in red) next to a second route (in blue), the latter being a straight line between the starting and finishing points. Which of the two is the shortest route?

Here is a second, seemingly banal question. Is Greenland the same size as Africa? If you look at the map below, it is very tempting to answer that they are about the same size. Note also the phenomenal size of Antarctica, compared with the other continents.

A simple look at the British Encyclopedia website tells us that Greenland has an area of 2,166,086 square kilometers, while Africa has an area of 30,365,000 square kilometers (Britannica, n.d.-1-2). Africa is actually 15 times bigger! The projection over-represents the Danish by a factor of 15.

The British encyclopedia also tells us that Antarctica has a surface area of 14.2 million square kilometers, half that of Africa (Britannica, n.d.-3). The projection clearly suggests otherwise.

Returning to *Jean-du-Sud’*s route, the curved path is the shortest of the two. This answer goes against popular wisdom to the effect that “the shortest way is a straight line”.

The adage isn’t wrong, the map is… and perhaps our notion of geometry. It is impossible to go “in a straight line” by following the surface of the earth (a sphere). So on or usual map representation, surfaces are wrong and straight lines are long as well. What kind of sorcery is this?

None at all. Or rather, its mathematics.

## Projections

The surface of the earth is the surface of a sphere. Two dimensions are needed to uniquely identify a point on its surface. The accepted coordinate system designates a point by its longitude and latitude. For example, Quebec City is located at 6°49’17.6 “N, 071°12’13.8 “W. It’s a two-dimensional surface. A map is also a two-dimensional surface.

We can therefore take every point on the earth’s surface and project it on a map while respecting the most basic property of all: that neighboring points on earth should remain neighbors on the map. In mathematics, this transformation exercise is called a **projection**.

So far, so good. It’s perfectly possible to project earth’s surface onto a map. It has been done done four times since the beginning of this post! That said, we would like to be a little more demanding of our projection, so as to avoid the problems of curvature of “straight lines” and the erroneous surface representations. To do this, we should consider two restrictions on a projection.

Firstly, it should not distort angles, i.e. a given angle on the earth’s surface should translate into the same angle as measured on the map. A projection with this property is called **conformal**. Secondly, a projection should also respect the proportionality of surfaces, i.e. it should respect relative sizes. A projection with this property is deemed to **preserve surfaces**.

In 1837, in his “Remarkable Theorem”, Gauss developed differential geometry results that proved, in essence, that a projection of the earth cannot simultaneously preserve surfaces and be conformal. No map is perfect… even those we have not yet found! Any projection of the earth will distort either the angles, or the surfaces, or both! So Gauss ended the quest for a perfect earth projection: there will always be distortions.

## A Few Examples

The best way to convince yourself of this fact is to examine different projections and look at what they do to angles and surfaces. To illustrate the effect of a projection, let us first glue circles of equal size to its surface (image below). If a projection distorts the angles, the circles will become ellipses, ovoids or any other distorted surface. If a projection distorts surfaces, some circles will become larger and others smaller.

### The Mercator Projection

The Mercator projection is well known to everyone. It’s THE projection that is used by every map of the world. Here’s what the circles look like under this projection.

Note that the circles are still circles (thus preserving angles), but that they are significantly larger at the poles than at the equator. This observation reconciles the encyclopedic data with this representation of the world: it is now understandable that the surfaces are exaggerated for Greenland and Antarctica.

This result is well known to navigators, as it implies that the distance scale is not the same according to the navigation latitude. This is why nautical miles are measured on the latitude axis (and not the longitude axis).

The Mercator projection is well suited for navigation, as it doesn’t distort angles, so it’s easy to identify bearings as they are the same on a compass and on a map (saved for declination/variation). There is furthermore not much distortions at the equator. The projection was invented by Gerardus Mercator (rendition in the first image) at a time when people were trying to cross the Atlantic efficiently, which was done close to the equator.

### The Equal Area Cylindric Projection

Is it possible to build a projection that preserves surfaces? Yes, but it will distort the angles.

Here, the circles are distorted, but if we were to do a thorough check, we would find that they all have the same surface. Note that Africa has recovered its rightful size in comparison to Greenland. Similarly, Antarctica has lost a lot of projected space!

### The Gnomonic Projection

No, it is not a bad psychedelic movie from the ’70s. This type of map actually exists! In fact, it is a favorite for ocean crossings, as it has the property of (locally) representing the shortest paths with straight lines. On the other hand, it distorts angles and surfaces, particularly outside its “center”. This kind of projection is therefore useful for small regions of the earth (such has an ocean), but does poorly as a “general map”.

## Conclusion

We could be here all day. The software I use to draw projections gives me the embarrassment of more than thirty projection types. In all cases, Gauss has shown us that none of them will both respect angles and surfaces.

In the film “The Matrix”, Morpheus introduces Neo to the matrix by revealing that his universe is a lie. Of course, the film is a reworking of the allegory of the cave, but now that we know that our maps also distort reality, the “first matrix” trophy may go to Gerardus Mercator. Maps are not what they seem.

## References

Britannica.com (s.d.-1). *Africa*, web page retrived in December 2023 at this address.

_____________ (s.d.-2). *Greenland Summary*, web page retrived in December 2023 at this address.

_____________ (s.d.-3). *Antarctica*, web page retrived in December 2023 at this address.

Wikipedia (s.d.). *Theorema egregium*, web page retrived in December 2023 at this address.

________ (s.d.). *Indicatrice de Tissot*, web page retrived in December 2023 at this address.