On Conformality

I teach cartography for a living, at UW-Madison, and last week I spent some time lecturing about projections. I think that this is probably the most difficult topic to teach — it’s the most technical and abstract, and we tend to avoid math, even though the topic is entirely mathematical in nature. Many students do not have the necessary background to delve into the equations and transformations.

Eventually, it comes time for students to learn about conformal map projections. Several online resources, and even some textbooks¹ tell readers that a conformal map projection preserves shapes. It’s what I was taught. It’s a nice and simple way to understand conformality with wading into the messy and confusing mathematics behind it.

But it’s also entirely false. If we could keep shapes perfectly preserved as we went from globe to map, we’d have no distortions at all. In reality, a conformal projection preserves local angles at infinitely small points. Now, that’s a rather abstract thing to consider, and so I can understand that an instructor would like to explain what that means in real-world terms to students. But saying that conformality preserves shape is misleading and confusing.

Here’s Greenland on three conformal projections:

Those three images are not the same shape. As an example, the little peninsula on the northwest coast (where the town of Qaanaaq is located), changes size and position relative to the rest of the island. Since conformal projections do not preserve areas, different parts of Greenland are being sized differently. If you take a polygon and inflate one part of it, it’s not the same shape anymore.

This is not mere pedantry; this language has actual negative effects on students. I’ve been a teaching assistant in a class where students were taught that conformal projections preserve shapes. Later, they did exercises where they visually assessed distortions on map projections. Several of them failed to correctly identify conformal projections because they saw changes in shapes like those in the example above. They reasoned, like I do, that those three things were not the same shape anymore, and so couldn’t be conformal based on what they had been taught. What they were heard in class conflicted with their experience, rather than being reinforced by it. This is a failure of the learning process.

So, what to tell them instead? Local angles are still hard to grasp, and don’t mean much in terms of looking at the big picture of the map. What I teach them is that conformality preserves the look of places on the earth, and I make clear that this doesn’t mean “shape.” “Look” is a fuzzy concept, but some visual examples help reinforce it — compare the three Greenland images above to two images on non-conformal projections:

The conformal ones, while a different shape, have a lot more in common with each other than the two non-conformal ones. The example I give in class is that rectangles and squares both have a similar look (and have the same angular relationships), even though they are different shapes. A triangle, though has a different look than either.

I do not understand why we persist in teaching that conformality preserves shape. Shape is a wonderfully concrete word, versus my own slightly vaguer alternative. It’s easy say shape, but it’s also wrong, and it quickly falls apart once the students spend ten minutes playing around with projections.

Perhaps I’m off base in my assessment, a fact which I partly suspect because I seem to be very much in the minority in avoiding the word “shape,” when so many respected cartographers make use of it. If you can set me right, I should be interested to hear a counterargument.

¹Muehrcke, et. al. is a textbook I have recently seen refer to conformality as shape-preserving, though that edition was a couple of years old, so the latest may have changed language. Likewise, the 1993 edition of Dent also appears to refer to conformal as shape-preserving, though I can’t speak for the most recent edition. Slocum, et. al., to their credit, make a point of explaining that conformal does not mean shapes are preserved. Robinson, et. al., do, too, but not quite as strongly. If you’ve got access to another textbook (or a more recent edition of Dent or Muehrcke), I’d be interested to hear what you find.