# 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.

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
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.

```

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