Map projections and distortion
Converting
a sphere to a flat surface results in distortion. This is the most profound
single fact about map projections—they distort the world—a fact that you will
investigate in more detail in Module 4, Understanding and Controlling
Distortion.
Imagine a
map projection as an attempt to reconstruct your face in two dimensions. Some
maps will get the shapes of all your features just right, but not the
sizes—your forehead and chin, for instance, may come out huge. Other maps will
get the sizes right, but the shapes will be stretched—maybe your full, round
mouth will appear wide, thin, and rather mean.
Some maps
preserve distances. Measurements from the tip of your nose to your chin, ears,
and eyes will be right, even though the size and shape of your features is wrong.
Other maps preserve direction. Your features may look weird, and they may be
scrunched up or set too far apart, but their relative positions will be
correct.
Finally,
some maps are compromises—they get nothing exactly right but nothing too far
wrong. In particular, compromise projections try to balance shape and
area distortion.
So the four
spatial properties subject to distortion in a projection are:
·
Shape
·
Area
·
Distance
·
Direction
Shape
If a map preserves shape, then feature outlines (like
country boundaries) look the same on the map as they do on the earth. A map
that preserves shape is conformal. Even on a conformal map, shapes are a
bit distorted for very large areas, like continents.
A conformal
map distorts area—most features are depicted too large or too small. The amount
of distortion, however, is regular along some lines in the map. For example, it
may be constant along any given parallel. This would mean that features lying
on the 20th parallel are equally distorted, features on the 40th parallel are
equally distorted (but differently from those on the 20th parallel), and so on.
Area
If a map preserves area, then the size of a feature on
a map is the same relative to its size on the earth. For example, on an equalarea
world map,
To look at
it another way, a coin moved to different spots on the map represents the same
amount of actual ground no matter where you put it.
In an
equalarea map, the shapes of most features are distorted. No map can preserve
both shape and area for the whole world, although some come close over sizeable
regions.
Distance
If a line from a to b on a map is the
same distance (accounting for scale) that it is on the earth, then the map line
has true scale. No map has true scale everywhere, but most maps have at least
one or two lines of true scale.
An equidistant
map is one that preserves true scale for all straight lines passing through
a single, specified point. For example, in an equidistant map centered on
Direction
Direction, or azimuth, is measured in degrees of angle from north. On
the earth, this means that the direction from a to
b is the angle between the meridian on which a lies and the great
circle arc connecting a to b.
The azimuth of a to b is 22 degrees.
If the azimuth value from a to b is the same on a map as on the earth, then the map preserves direction from a to b. An azimuthal projection is one that preserves direction for all straight lines passing through a single, specified point. No map has true direction everywhere.
A few
projections with different properties. The Lambert Conformal Conic preserves
shape. The Mollweide preserves area. (Compare the
relative sizes of
More about scale
Scale is
the relationship between distance on a map or globe and distance on the earth.
Suppose you
have a globe that is 40 million times smaller than the earth. Its scale is
1:40,000,000. Any line you measure on this globe—no matter how long or in which
direction—will be
Now suppose
you have a flat map that is 40 million times smaller than the earth. (See the
problem coming? Instead of comparing a big orange to a little orange, we're
comparing a big orange to a little wafer.) This map also has a scale of 1:40,000,000,
but because the map and the earth are differently shaped, this scale cannot be
true for every line on the map.
The stated scale of a map is true for certain lines only. Which lines these are depends on the projection and even on particular settings within a projection. We'll come back to this subject in Module 4, Understanding and Controlling Distortion.
Not all of the earth's curves can be represented as straight lines at the same fixed scale. Some lines must be shortened (and others lengthened).
Expressing
map scale
There are three common ways to express map scale:
Linear
scales
Linear scales are lines or bars drawn on a map with
realworld distances marked on them. To determine the realworld size of a map
feature, you measure it on the map with a ruler or a piece of string. Then you
compare the feature's length on the string to the scale bar.
A typical scale bar.
Verbal
scales
Verbal scales are statements of equivalent distances. For example, if a 4.8
kilometer road is drawn as a 20 centimeter line on a map, a verbal scale would
be “20cm = 4.8km.” You could also formulate the scale (reducing both sides by
20) as “1cm = .24km.”
Representative
fractions
Representative fractions express scale as a fraction or ratio of map distance
to ground distance. For example, a scale of 1:24,000 (also written 1/24,000)
means that one unit on the map is equal to 24,000 of the same units on the
earth. Since the scale is a ratio, it doesn't matter what the units are. You
can interpret it as 1 meter = 24,000 meters, 1 mile = 24,000 miles, or 1 hand
width = 24,000 hand widths (as long as it's the same hand).
Small
scale and large scale maps
It's easy to mix these terms up. Here's one way to
keep them straight: on a largescale map, the earth is large (so not very much
of it fits on the map). On a smallscale map, the earth is small (so all or
most of it fits on the map). A map of your town, or your property, is going to
be a largescale map. A continental or world map is a smallscale map.
Another way to think of the difference in terms of representative fractions. The larger the fraction, the larger the map's scale. For example, 1/10,000 is a larger fraction than 1/1,000,000. So a 1:10,000 map is larger scale than a 1:1,000,000 map.
Measuring distortion using Tissot's Indicatrix
In the
nineteenth century, Nicolas Auguste Tissot developed a method to analyze map projection
distortion.
An
infinitely small circle on the earth's surface will be projected as an
infinitely small ellipse on any given map projection. The resulting ellipse of
distortion, or indicatrix, shows the amount and type
of distortion at the location of the ellipse.
For example, if an indicatrix is elongated from north to south, shape is correspondingly distorted at that location on the map. The same goes for east–west stretching or oblique stretching. On a conformal map, the indicatrices are all circles, but they vary in size. On an equal area projection, the indicatrices have varying ellipticity, but the same area.

The Mercator projection is conformal. The
Sinusoidal projection preserves area. The
EqualArea Cylindrical projection In the
Robinson projection, neither shape 
Tissot indicatrices for four projections.