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.

16 thoughts on “On Conformality

  1. Thanks for this post! I had this question when we were going over projections in my GIS class the other day. Maybe people need to take more math classes so that we can talk about these kind of subjects in a more concrete manner, but hey that is just my opinion.

    1. I’m happy to help out! I think that more math would certainly be handy in understanding these things, but I wonder if it might not form too big of a barrier as far as getting people into the field goes.

  2. Great post. I have had the same issue, and have resorted in the past to qualifying phrases like “it generally maintains shape, but really at only infinitely small points”. Then in the past couple of years I have put in a section on Tissot’s Indicatrix, based on the section and figures in Slocum (3rd ed.) which seems to clarify things a lot for my students.

  3. Hi Daniel, followed you over here from Cartastrophe, great blog (both actually!). Good write up on conformal projections. I must admit I have never given this much thought. I guess I always understood ‘shape’ in this context to mean that it remained easily recognizable which is how I would define your use of ‘look’ as well. But you are right, it can be misleading if not well explained. I don’t often have to discuss projections in polite conversation ; ) but if I do I’ll make sure to emphasize the ‘look’ aspect of conformal projections.

    1. David, so glad you could join me here as well. I don’t think I gave the issue much thought, either, at first. When I learned it as shape, I eventually just adjusted my understanding of what “shape” meant in that context and stopped thinking about it so literally. I didn’t really realize the magnitude of the problem until I saw how some students struggled with it.

    1. Thanks so much for sending this along — hadn’t seen that article. I have certainly bought into the whole “Mercator Effect” line up to this point. I think that their sample is a bit limited (it seems like nearly all cartography studies are based on samples of college students and then use that population to make sweeping general claims), but they may well be on to something.

  4. Hi Daniel,

    Great post about conformal projections. I also teach GIS, and of course must go through projections. So I’ve come across the same dilemma in making these concepts sink in without showing the math. Have you been able to introduce some crude math concepts related to this subject matter, or found other resources to assist with this?

    1. Hi Greg,

      I use the model of developable surface/reference globe as a stand-in for the math. I make it clear to them that this is just a model, and that there are projections that cannot entirely be conceived as fitting with it (such as anything pseudocylindrical). It seems to work reasonably well with them. We make them go through some projection exercises in lab — choosing a projection and parameters for different situations, and for the most part they seem to get it well enough to be able to go on and make great maps. But we don’t really get much more complicated than the model (though I tell them that if they have the math background, they can read up on the details of different projections online).

  5. Daniel,
    I entirely agree that ‘look’ is a fuzzy concept, which leads me to make one comment (entirely aimed to be constructive so please don’t take this as criticism or pedantry):
    The 2 images of Greenland: they don’t ‘look’ to me as if they are actually equal in area! (meaning the number of pixels that make up the image. I’m guessing that (if I’m right) this is an issue of web page space rather than an optical illusion. If you use these images in class is the Cylindrical image bigger than it appears here?

    I just know that my students would have been saying “but Chris those 2 images don’t look to be the same in area!”

    So I guess my point is it’s a related message about getting the point across. the concept of equal area needs to be supported by images that look equal.

    A minor point in an excellent post, but one that I hope is useful, particularly for students who will no doubt rely upon your posts for clarification and discussion.

  6. I sometimes think that we start to learn about projections in the wrong way– by looking at maps. We should start by looking at a globe. Every year I teach a science class at a friend’s 4th grade class room. I have found that they grasp the concept of projections easier by going a bit backwards. I take a basketball and ask someone to wrap it as neatly as possible as a present. It is hard but it gets 25 4th graders engaged really quickly. I then show them a globe, and how they can peel away parts of it to make flat. They can then see how difficult it is to take a sphere and represent it on a flat surface. Once they understand that, we can talk about the different high level types of projections and how they can be useful. I *think* this works for 4th graders, but they have much more fun when we then do an exercise using layers of maps and the best way to get from A to B, and how that changes as layers are added representing water, topography, bridges, etc.. (but I digress).

    Projections are the hardest thing to understand in mapping, and the novice user typically has no clue and everyone things Greenland looks much different that it is. Personally, I think that globes should be used much more in schools than paper maps, but that would require a lot of change….

    Perhaps a globe is a better way to start with when teaching projections– start with the original sphere and then move to the map, rather than starting with the different shapes and types of projections. Once the basic concept is understood, the context is set for a better understanding of the math reqired to flatten the earth.

    1. I quite agree that a globe is the way to go with projections. I start out my class by talking about some basic geodesy, and only once we’re set on the earth as a 3d object do we move on to how it’s flattened. I use a couple of models for that — first off I have them think of peeling an orange, and how it’s impossible to get it to lay flat without distorting it. And then I use the model of the reference globe/developable surface, in which I liken projections to photographs of the earth, with the photo paper and the light source being movable, which produces different projections. It’s a limited model, in that only a handful of projections make sense on it, but it seems to help get them started (and I clarify to them that it’s just a model and won’t always work).

      I like the basketball-wrapping idea. I think that’s definitely a good way to start things, too.

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