B. Projections are the ways cartographers use to juggle the properties of a map, so as to optimize the ones they need for the purposes to which their map will be put. Emphasizing certain properties at the expense of others is done through the art of projection. There are four broad categories of projection, each with a number of subcategories. 1. Azimuthal projections (sometimes called planar or zenithal projections). Imagine a globe made up entirely of wires, wires representing various latitude and longitude lines. Now imagine lighting up this round wire cage and placing a sheet of paper against it, so that the paper is tangential at one spot on the cage. You'd turn on the light and then trace the shadows of the wires onto the paper. Then, you would have a grid on which you could draw in the continents and whatever you wanted to show. This is the idea behind an azimuthal projection. It's called that because any straight line drawn from the point of tangency (where the paper touched the cage) is a true great circle. There are a few types of azimuthal projections, that differ from one another in where you put that light bulb. a. A gnomonic projection results when the light is placed in the center of the wire cage. Usually the map is tangential to one of the poles. If so, this map shows meridians as azimuths (true compass directions) and parallels as concentric circles that are spaced farther and farther apart as you move away from the pole. This map distorts area and shape, especially as you approach the margin. It does have the interesting property that any straight line drawn on it happens to be a great circle route. b. A stereoscopic azimuthal projection results when the light is placed on the antipode from the point of tangency. The point of tangency is usually one pole, so the light would be placed at the other pole. This map looks a lot like the gnomonic one, except the parallels stay the same distance apart as you move towards the margin. Meridians are still azimuths, but you lose the great circle property of other straight lines. A strength of this projection is that shape is true: This is a conformal projection. Area, however, remains badly distorted (but at least the distortion is concentric). c. An orthographic azimuthal projection results from placing the light at infinity (well, at least in your mind's eye!). The light rays, then, come to the wire cage parallel to one another. This means that the concentric parallels are projected as closer and closer to one another as you approach the perimeter, which creates an image of the earth that looks a lot like it would if you were really in outer space gazing at it. The maps look pretty cool, then, but they do distort both shape and area: They are neither equivalent nor conformal. 2. Conic projections are created by placing a light in the center of the wire ball and then setting one or more paper cones on one of the poles of the wire globe, as though it were wearing a hat or dunce cap! The result is a semi-circular map. Where azimuthal projections are tangent only at one point (commonly, though not always, a pole), a conic projection is tangent all along a line, a parallel called its "standard parallel." The standard parallel possesses true scale, shape, and area: The map is equidistant, conformal, and equivalent along the standard parallel. The lucky parallel is determined by the angle of the cone. If it's a low angle cone, the projection is tangent at a high latitude; if it's a high angle cone (like a dunce cap or witch's hat), the map is tangent at a lower latitude. These maps are best suited to mapping the mid-latitudes. The US is very commonly shown on a conic projection and generations of schoolchildren looking at these maps think that Maine and Washington State contain the northernmost reaches of the contiguous 48 states (when in fact that honor belongs to Minnesota, with its Lake of the Woods!). Meridians show up as straight lines radiating from the middle of the top edge, while parallels show up as concentric semi-circles spaced farther apart as you move away from the standard parallel(s). This means scale, shape, and area become more and more distorted away from the standard parallel(s). So, overall, these maps are neither conformal nor equal area, but they're a sort of compromise between the two. There are several variants on this basic idea. a. A tangential conic projection is basically just what I got done describing: There is one paper cone of one or another steepness and one standard parallel at higher or lower mid-latitudes. b. A secant conic projection takes a bit more imagination. It involved a "paper" cone seated so that it passes "through" the wire globe along TWO standard parallels. This way, you get two error-free parallels and the relatively small band in between them has only minimal distortion. Raw secant projections are neither conformal nor equivalent, becoming more and more distorted as you move away to the north and south from the two standard parallels. One of the problems is that the meridians are pulled apart a bit more than they should be, because they are shown converging on the tip of the cone, rather than the pole of the globe far below it. It is possible to manipulate secant conic projections so that they become either conformal (true shape) or equivalent (true area), and here are two examples: i. Lambert's conformal conic projection pulls the parallels apart a bit to compensate for the exaggerated separation of the meridians at higher latitudes as they approach the tip of the cone (rather than the pole of the globe). ii. Albers' equal area conic projection entails pushing the parallels closer in to one another to compensate for the increased area created by those stretched-out meridians. c. A polyconic projection is another exercise in imagination. You imagine a wire globe with several cones of paper on it, each one of a different steepness, and all superimposed. The idea is to draw the map around each one's standard parallel and then reconciling the various maps. It preserves true scale among the several standard parallels, but never achieve conformality or area equivalence: Such maps are a compromise between these two virtues and, like the other conic projections, are best suited to the mid-latitudes. 3. Cylindrical projections are based on the idea of wrapping a roll of paper around the wire globe, putting a light in the center, and tracing the grid onto the paper roll. Most versions of this kind of map are tangential along the equator, though there are some newer versions tangential along meridians. This yields a nice, rectilinear map. Parallels are straight lines and so are meridians, and they cross at right angles. This kind of map typically shows the whole world, except those latitudes very close to the poles. These maps are intuitive for most people to read, but they do grossly distort size and shape of landmasses and water bodies. The meridians are primarily at fault here: In the real world, they converge to the poles; here they are parallel to one another and do not converge at all. The higher latitudes are strongly distorted in shape in an east-west direction, and they are also grossly bloated in area. a. The most famous variation on this map is the Mercator projection. This projection has artificially had the parallels pulled even further apart than the shadows would indicate to compensate for the distortion in the meridians. So, this creates the absurdity of seeing Greenland as larger than Africa or South America, when it is much smaller. This is not an equal area map by a long shot! It is, however, conformal and is incredibly useful to navigation, the purpose for which it was published in 1569! Any straight line drawn on this map is a true compass heading (or "loxodrome" or "rhumb line," if you want to get fancy, or line of constant compass direction). That means, if you drew a straight line from Place A to Place B and then measured the angle at which your line crosses the meridians, you could just point your boat at that angle and sail on. This route will get you there with the least navigational fuss, though it won't be the shortest path there (which is the great circle route and using the great circle route requires adjusting your heading from time to time). This property of giving you a single heading you can use for your trip is why this map remains the most widely used global navigation chart. b. An historian named Arno Peters developed a "perfect" map in a press conference back in 1973, the Peters Projection. He claimed that the Mercator map was "racist," because it made Africa and South America look small in comparison with Europe, North America, and Greenland. Yes, the Mercator projection does do that, but the purpose of the map never was to show equal area; it was to aid navigation! Basically, this boils down to Peters saying that, if you're the captain of a boat and you'd like an easy to interpret navigation map, you're racist. This whole controversy was raised by an historian with less training in map projections than you now possess, who was unaware that there are boodles of equal area map projections out there and in common use! So, he came up with an equal area cylindrical projection, which comically distorts shape pretty much everywhere on the map. Africa, for example, in the real world is roughly as wide east-west as it is north-south, but on the Peters map it comes out as being twice as long north-south as it east-west. Compare the Peters map with a few other equal area maps, for which I thank Peter H. Dana's The Geographer's Craft Project. In short, this is really an absurd controversy for any geographer or anyone else with any exposure at all to map projections and cartography. There are many superior equal area projections out there, and they should be used for any map showing distributions of such things as population and wealth that are politically loaded. The best thing we can come away with from this is that, apparently, a number of other people, knowing as little about projections as Arno Peters, were themselves happily plotting such distributions on a navigation map that is inappropriate to use for showing distributions in which area is a key consideration. 4. Mathematical projections are not really projections in the sense of "projecting" a shadow design from a wire cage onto a piece of paper. They are arrived at with various mathematical functions and can even get thoroughly bizarre, such as those oddball speculations on what the earth would look like as a cube, star, dumbell, and such. a. Non-projected cylindrical maps are produced by just specifying that meridians and parallels will be exactly the same distance apart and cross one another at right angles to form a perfect grid. Thanks to Peter H. Dana's The Geographer's Craft Project for this image. b. The Mollweide's homolographic projection is pretty simple: Draw a straight horizontal line to stand for the Equator. Put a vertical line half its length through the middle of the horizontal line and let it stand for the Prime Meridian. The 2:1 ratio is because the Equator is a full great circle and any meridian is half a great circle. Connect both ends of the Equator line to both ends of the Prime Meridian line with an ellipse, which stands for the antipodal meridian (180°) on either side of the map. Now fill in the other meridians at equal distances from one another, creating less and less extreme ellipses as you work back toward the Prime Meridian, making sure they all touch at the North Pole and the South Pole. Fill in selected other parallels as straight lines above and below the Equator, making them slightly closer together as you approach the poles. You now have a framework on which you can fill in the entire world and create an equal area map, which also preserves true scale and shape along the Equator and Prime Meridian. Thanks to Peter H. Dana's The Geographer's Craft Project for this image. c. The Sanson-Flamsteed sinusoidal projection is exactly like the Mollweide, except, instead of ellipses, you use sine curves. This one is also an equal area projection, but it creates less distortion in the tropics at the expense of more distortion in the polar regions than the Mollweide. Thanks to Peter H. Dana's The Geographer's Craft Project for this image. d. Wouldn't it be nice to create an equal area map that has Mollweide's virtues in the polar regions and Sanson-Flamsteed's in the tropics? Attempts have been made to cut and paste the two together, and the most famous one, the basis of the Goode's World Atlas maps, is copyrighted, so I won't show it directly here, but I'll send you to a page that shows it. It's the Goode's interrupted homolosine projection. It consists of a Mollweide projection above 40° N or S and a sinusoidal in the areas from 40°N to 40°S. Also, Goode tore the earth into two gores in the Northern Hemisphere, each with its own vertical central meridian. In the Southern Hemisphere, there are four gores, again each centered on its own vertical meridian. He went crazy with all these vertical "central" meridians, because he knew the shape distortions were least along the Equator and central meridian of the parent projections, so he decided to straighten out a few more and create more areas with little distortion. Straightening out the selected central meridians meant the earth is torn into this weird cut-out shape, but you can make the tears where they are least disruptive to the purpose of the map (e.g., you can put them in the oceans for any maps about distributions on land). This is probably the best of the equal area maps in preserving equivalency while doing the least harm to conformality. This map makes the Peters projection controversy look even sillier: Goode's. e. The Robinson Projection is a good compromise projection. It distorts all areas and shapes a little bit, but overall distortion is minimal (except around the edges). It is also a very attractive map without the weirdazoid torn gores of the Goode's. It shows parallels and the central meridian as straight lines and then uses a table of longitude coördinates put together by Arthur H. Robinson (folk hero in cartography) in 1963 for every 5° of latitude. You look these up and put a dot on that parallel in the right place and then interpolate between them to make the map. It is very popular among atlas and mapping companies and the National Geographic uses it beaucoup. Again, I thank Peter H. Dana for this map: Quick study guide before the next lecture: Now, I know you're all pretty bewildered by this extreme variety of map projections and getting a little scared about how I would test you on them. What's really scary is I've only given you the barest idea of the possibilities here! I do not expect you to memorize all the attributes of each projection type here. What I want you to do is remember that there are several broad categories of projection: azimuthal, conic, cylindrical, and mathematical. You should remember how each of the basic types is created: a wire globe with a paper tangent at one point (azimuthal); a wire globe with a paper cone or cones sitting on it or actually cutting through it (conic); a wire globe with a cylinder of paper wrapped around it (cylindrical); and various artificial grids (mathematical). You should also know in general which ones are best for showing areas in the mid-latitudes (conic), the polar areas (azimuthal), the whole world (cylindrical or mathematical). Also, be aware of the construction and original purpose of the Mercator's projection and the nature of the Peters projection critique and why the Peters' projection controversy is so silly to geographers, cartographers, and anyone else with elementary training in map projection. You also need to focus on the basic properties of maps (equivalency, conformality, equidistance, and true direction) and why the act of projecting a round earth on a flat sheet of paper means that you can't preserve all properties at once. Of these, know that equivalency (equal area) and conformality (true shape) are the most diametrically opposed map properties: It is impossible to have both of these together. If all these maps and the art of projection hit you as kind of cool, actually, you can explore allllllll sorts of projections at the following sample of sites: Picture gallery Peter H. Dana's The Geographer's Craft Eric Weisstein's World of Mathematics Matt Rosenberg's Cartography On to the next lecture, on map scale issues.
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First placed on web: 09/16/00 Last revised: 06/08/07