Geography 140
Introduction to Physical Geography

Lecture: Map Projections

     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.

              [ Azimuthal gnomonic projection ]

              [ Azimuthal gnomonic map ]

           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).

              [ Azimuthal stereographic projection ]

           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.  

              [ Azimuthal orthographic projection ]

              [ Azimuthal orthographic map ]

        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.

              [ Tangential conic projection ]

           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. 

              [ Secant conic projection ]

              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).

                   [ Lambert's conformal conic map ]

               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.

                   [ Albers' equal area conic map ]

           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.

              [ Secant conic projection ]

        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.

              [ Mercator projection ]

           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 

              [ Peters projection ]

              Compare the Peters map with a few other equal area maps,
              for which I thank Peter H. Dana's The Geographer's Craft 

              [ Eckert IV equal area projection ]

              [ Robinson sinusoidal equal area projection ]

              [ Behrmann cylindrical equal area projection ]

              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 
           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.

              [ unprojected cylindrical grid ]

           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.

              [ Mollweide homolographic map ]

           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.

              [ Mollweide homolographic map ]

           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:   

           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:

              [ Mollweide homolographic 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 
   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.


Document and © maintained by Dr. Rodrigue
First placed on web: 09/16/00
Last revised: 06/08/07