Distortion comes in two primary forms: deformation of angles, and variation in relative area measure. Every map has at least one of those two primary forms. Many maps have both. One might choose to measure or characterize other things, such as “scale”, distances, directions, or bearings, and then measure how well a map preserves one of them. However, ultimately, any other metric that you choose arises out of the inevitable distortion of angles, relative areas, or both. Those two metrics are irreducible.

I will carefully explain these two forms of distortion. As described by Nicolas Auguste Tissot, 1881, in

*Memoire sur la representation des surfaces et les projections des cartes geographiques*, you can characterize the distortion at a point as the abuse an infinitesimal circle from the sphere suffers when projected onto the plane. Because the circle is infinitesimal, only two things can happen to it: It can gain or lose area, and it can get squashed into an ellipse. Here I portray the original circle on the left and an example projection on the right: You can imagine an unlimited number of crosses in the original circle. I show four, each in a different color. After projecting, most of the crosses can end up no longer meeting at right angles, but there will always be one that does, or possibly all of them will still. Unless all of them remain at right angles, there will always be one axis with the greatest change in angle. Among those axes shown, that would be the green one. If all of the axes keep their right angles, then there is no angular deformation at that point. The ellipse generated by this mechanism is called the

*Tissot indicatrix*.

Secondly, the area has changed in this example. The ratio of the area of the resultant ellipse to the original circle is the amount of area inflation or deflation at the point. If the area does not change, then there is no “distortion” of area at the point, regardless of how much angular deformation there might be. Across a map, these measures change point-by-point. If all the ellipses have the same area, then the map is equal-area. If all of the ellipses are actually circles, and therefore show no angular deformation, then the map is conformal. A map cannot be both conformal and equal-area. Even so, specific points, lines, or paths can be without distortion as long as the points do not form a region, however small.

Here is a map showing the Tissot indicatrix for an entire map, spaced at 15° intervals: This map is neither equal-area nor conformal: Its Tissot indicatrices vary in area and most of them are ellipses.

By contrast, here is an equal-area map’s Tissot indicatrices: All ellipses have the same area, but they show varying amounts of angular deformation.

And this is a conformal map: All indicatrices are circles, but their areas vary.

“Variation in relative area measure” is cumbersome. To express the concept, the literature talks about “area distortion” or “areal distortion”. I dislike these terms because I have found them to spark misunderstandings in people’s grasp of projection distortion. Instead, I use

*flation*, which means “area inflation or deflation”. I introduced this term in Battersby, Strebe, and Finn, 2016,

*Shapes on a plane: evaluating the impact of projection distortion on spatial binning*.

It might seem tempting to think of angular deformation and flation as “complementary”. In a sense, they are, because together they represent everything there is to say about distortion at a point. However, they behave quite distinctly.

On the one hand, flation is relative. The scale you assign to a map is arbitrary, and so if the relative sizes of circles on a conformal map differ, who is to say which size is “correct”? You are to say… but your choice is arbitrary. All you can meaningfully talk about is the

*ratio*of areas, and how that varies, across the map.

On the other hand, angular deformation is absolute. It does not matter what scale you assign to a map, or how you shrink or enlarge it, the ellipses still have the same proportions, and they represent the same amount of angular deformation.

Because of this very different semantic of relative flation and absolute deformation, you cannot meaningfully compare the two kinds of distortion. Let’s say, for example, that we wish to show how distortion is distributed across a map. Because the map is neither equal-area nor conformal, we can show where, and how much, angular deformation there is, and where, and how much, flation there is, using different colors. We’ll use green for flation and magenta for angular deformation. The deeper the color, the more distortion. I “bin” the colors into distinct bands, rather than show distortion in continuous tones, because it is very hard to compare color shades of regions that are distant from together, otherwise. Left is the map projected at its “nominal scale”. Right is the map scaled with respect to its “nominal scale”. Notice there is no difference in the distortion pattern.

Now let’s see what happens with flation: Left is the map projected at its “nominal scale”. Right is the map scaled with respect to its “nominal scale”. Notice the distortion pattern changes considerably. That’s because we are measuring the distortion in absolute terms against the nominal scale that we declared. In the end, the Tissot indicatrix only knows what you tell it about the size of the globe you projected from.

An interesting characteristic of projection distortion is that it is invariant against coordinate rotation. What is coordinate rotation? That’s when you grab the globe’s axis and yank the north pole to some other place. When you project using the same formulæ, you get the same outline for the map, but the contents of the map are all shifted. Neverthless, the distortion of each point

*in the map*, and the patterns of distortion across the map, do not change. For example: Lo! No difference in distortion even when you move a different point to the center: Another interesting characteristic of projection distortion is that the angle formed by meridians and parallels tells you nothing about the distortion. It is true that a conformal projection always has perpendicular meridians and parallels, and this is true even under coordinate rotation. However, if the map is •not• conformal, then even if it had perpendicular meridians and parallels in one view, it would not after coordinate rotation.

So far we’ve discussed distortion as a phenomenon of points. That’s because the Tissot metric applies to individual points. What if we want to compare two different maps to ask, “Which one has less distortion?”

This is a hard problem. Many different schemes have appeared in the literature. Each has its merits. Each has its deficits. Each ranks the overall distortion of a selection of maps differently.

The problem starts out hard because distortion, even at the level of a point, consists of two independent, incomparable phenomena: angular deformation and flation. Without making arbitrary decisions about how to go about it, you cannot claim that this much angular deformation is worth that much flation. If you cannot do this at the level of a single point, how do you do it across the unlimited points of an entire map?

Nevertheless, we really seem to want answers about which projection is better, so we keep trying.

Interestingly, the question of which

*conformal*map is best has a rigorous and beautiful answer. First some preparatory remarks. In the distortion maps I show above, you can see bands of distortion. Each band has edges, and you can think of an edge as a path. In the parlance of mathematical cartography, such a path is called an

*isocol*or

*isomegeth*or

*level curve of distortion*. An isocol has constant distortion measure along its length. In the case of conformal maps, this isocol is a path of constant flation. (Originally, isocol referred

*only*to the level curve of area measure on a conformal map, but in recent decades researchers have generalized usage to also include the path of constant flation on other maps, as well as the path of constant angular deformation on any map. Obviously, a conformal map will have no isocols of angular deformation, and an equal-area map will have no isocols of flation. A map that is neither equal-area nor conformal will have both, and on most maps, they will not coincide.)

In 1853, renowned mathematician Pafnuty Chebyshev developed a hypothesis about this problem of the optimal conformal map. First, you define what region you are interested in using a closed boundary path. When you do this, Chebyshev conjectured, the optimal conformal map for that region is the conformal map that has an isocol coincident with that boundary. In 1896, Chebyshev’s hypothesis was proved by D.A. Gravé (sometimes transliterated “Grawe”). In the century plus since, researchers have devised various optimal projections for specific regions of the world based on the Chebyshev criterion.

The reason Chebyshev’s mechanism works is because of some very deep and interesting properties of conformal maps. One of those properties is that, if you know the distortion characteristics of even just a short path in the conformal map, then you can know the

*entire conformal map*! This arises out of something called the continuation theorem. One of its implications is that every conformal map is unique, given a bounding isocol or any other path on it. Because it is unique, you can say

*the*map having an isocol that bounds the region of interest is

*the*one that is optimal for that region.

But optimal in what way? The thing about maps is that distortion can go out of control easily. A map that stretches the poles into lines, for example, has infinite distortion at those points, and distortion increases monotonically toward those poles, too, and so they are not just singularities that can be ignored; the entire region nearby is grossly distorted. This means that any measure of overall distortion that includes the poles will be heavily weighted by these most-distorted regions. What Chevyshev’s criterion does is to put a concrete limit on the distortion. If you have an isocol of flation delineating a region, then

*at no point inside that region will the flation ever exceed that of the bounding isocol’s*. The ratio of the flation of the isocol to the minimum flation within the region is

*the minimum it can possibly be*. It is by this simple, natural, and breathtakingly lucid criterion that conformal maps can be compared against each other. There are other ways to measure and compare flation, and by other metrics, how different projections ranking in distortion performance could change. However, with the power of Chebyshev’s criterion, little impetus for such research exists, and little has been done. There isn’t even any reason to compare. The goal is to formulate a projection with an isocol that bounds the region you care about. The closer your isocol is to your ideal region, the better you have met your goal.

What about a world map? With world maps, you have to be careful and state clearly what you mean: a world map that is only ripped at one point, like an azimuthal projection? A world map that is ripped along an entire meridian like the maps I show on this page? Something else? If you only interrupt at one point, the optimal conformal map is the stereographic, which is not very satisfying because it is infinite in extent. If you permit interruption along an entire meridian, like most world maps you see, then the optimal conformal map is the Eisenlohr.

In short, the theory of conformal projections is well developed. It draws upon the vast literature and flexibility of complex analysis.

Having disposed of this background material, I will next write a post, as time permits, about optimal equal-area maps.